1) The number of ways in which we can post 5 letters in 10 letter boxes is..
A) 50. B) 5¹⁰. C) 10⁵. D) none
2) The number of 5 digit telephone numbers having at least one of their digits repeated is...
A) 90000. B) 100000
C) 30240. D) 69760
3) A class has 30 students. The following prizes are to be awarded to the students of this class. First and second in mathematics. first and second in physics, first in chemistry and first in Biology. if N denote the number of ways in which this can be done, then
A) 400/N. B) 600/N
C) 8100/N
D) N is divisible by four distinct prime numbers.
4) A letter lock consisting of three rings marked with 15 different letters. if N denotes the number of ways in which it is possible to make unsuccessful attempt to open the lock, then
A) 482/N
B) N is product of three distinct prime numbers.
C) N is product of 4 distinct prime numbers. D) none
5) The value of ³⁵C₈+ ⁷ₖ₌₁∑ ⁴²⁻ᵏC₇ + ⁵ⱼ₌₁⁴⁷⁻ʲC₄₀₋ⱼ is
A) ⁴⁷C₇ B) ⁴⁷C₈ C) ⁴⁶C₇ D)⁴⁶C₈
8) If ᵐ⁺ⁿP₂= ᵐ⁻ⁿP₂ = 30, then (m,n) is given by
A) (7,3) B)(16,8) C)(9,2) D)(8,2)
9) Let p be a prime number such that p≥ 23. Let n= p ! + 1. The number of primes in the list n+1, n+2, n+3, ....., n+p-1 is...
A) p-1 B) 2 C) 1 D) none
10) The ten digit of 1! + 2! + 3!+ .... 49! is...
A) 1 B) 2 C) 3 D) 4
11) four dice are rolled. The number of possible outcomes in which at least one die shows 2 is.
A) 1296. B)625. C) 671. D) none
12) A set contains (2n+1) elements. The number of subsets of the set which contain an atmost n elements is.
A) 2ⁿ B) 2ⁿ⁺¹ C) 2ⁿ⁻¹ D) ²ⁿ
13) A is a set containing n elements. A subset P of A chosen. The set A is reconstructed by replacing the element of P. A subset Q of A is again choosen. The number of ways choosing P and Q so that P∩Q= ¥ is...
A) 2²ⁿ - ²ⁿCₙ B) 2ⁿ C) 2ⁿ- 1. D)3ⁿ
14) At an election, a voter may vote for any number of candidates not greater than the number to be chosen. There are 10 candidates and 5 members are to be chosen. The number of ways in which a voter may vote for at least one candidate is given by:
A) 637. B)638. C) 639. D) 640
15) The sum of all the five digits numbers that can be formed using the digits 1 2 3 4 and 5 (repetation of digits not allowed) is.
A) 360000 B) 660000
C) 366000 D) none
16) Eight chairs are numbered 1 to 8. Two women and three men wish to occupy one chair each. First the women to choose the chairs from amongst the chairs 1 to 4 and then the men select from the remaining chairs. The numbers of possible arrangements is ..
A) ⁶C₃.⁴C₂ B) ⁴C₂. ⁴P₃
C) ⁴P₂. ⁴P₃ D) NONE
17) The number of signals that can be generated by using 6 differently coloured flags, when any number of them may be hoisted at a time is..
A) 1956. B)1957. C)1958. D)1959
18) If the letters of the word RACHIT are arranged in all possible ways and these words are written out as in a dictionary, then the rank of the word RACHIT is..
A) 365. B) 481. C) 720. D) N
19) A five digit number divisible by 3 is to be formed using numerals 0, 1 ,2 ,3, 4 and 5 without repetition. The total number of ways in which this can be done is..
a) 216 B) 240 C) 600 D) 3125
20) The number of ways in which a mixed double game can be arranged from amongst 9 married couples if no husband and wife play in the same game is
A) 756. B) 1512. C) 3024. D) none
21) the sum of the divisors of 2⁵. 3⁷. 5³. 7² is
A) 2⁶. 3⁸. 5⁴. 7³.
B) 2⁶.3⁸.5⁴.7³ - 2.3.5.7
C) 2⁶. 3⁸. 5⁴. 7³ -1 D) none
22) If n objects are arranged in a row, then the number of ways of selecting three of these objects so that no two of them are next to each other is...
A) {(n-2)(n-3)(n-4)}/6 B) ⁿ⁻²C₃
B) ⁿ⁻³C ₃ + ⁿ⁻³C ₂ D) none
23) The number of ways of selecting 10 balls out of unlimited number of white , red ,blue and green balls is..
A) 270 B) 84. C) 286. D) 86
24) The number of non negative integeral solutions of x₁+ x₂ + x₃ + x₄ ≤ (Where n is a positive integer) is
A) ⁿ⁺³C₃ B) ⁿ⁺⁴C₄ C) ⁿ⁺⁵C₅ D) ⁿ⁺⁴Cₙ
25) In a certain test, there are n questions. In this test 2ⁿ⁻ᶦ students gave wrong answers to at least I questions. where I= 1,2,3...., n. If the total number of wrong answers given 2047. then n is equals to.
A)10. B) 11. C) 12. D) 13
26) The number of times the digit 3 will be written when listing the integers from 1 to 100 is
A) 269. B) 300. C) 271. D) 302
27) If n is a positive integer, the value of
E= (2n+1)ⁿC₀ +(2n-1)ⁿC₁ +(2n-3)ⁿC₂ +.....+ 1. ⁿCₙ is
A) (n+1)2ⁿ
B) f'(2) where f(x) = xⁿ⁺¹
C) 3ⁿ D) none
28) Number of positive integral solution of x₁x₂ x₃= 30 is ---
A) 24. B) 25. C) 26. D) 27
29) There are 3 piles of identical red, blue and green balls and each pile contains at least 10 balls. The number of ways of selecting 10 balls if twice as many red balls as green balls are to be selected is......
A) 4. B) 5. C) 6. D) 7
30) A set containing n elements. A subset of P of A is chosen. The set A is reconstructed by replacing the elements of P. A subset of Q of A is again chosen. The number of ways of choosing P and Q so that P∩Q contains exactly one element is .
A) n B) 3n C) n.3ⁿ D) n.3ⁿ⁻¹
31) The value of expression ᵏ⁻¹Cₖ₋₁+ ᵏCₖ₋₁ + .....+ ⁿ⁺ᵏ⁻²Cₖ₋₁=...
A) 0. B) 1. C) ⁿ⁺ᵏ⁻¹cₖ D) N
32) There are 5 different books on Mathematics, two different books on Chemistry and four different books on Physics. The number of ways of arranging these books on a shelf so that books of the same subject are stacked together is..
A) 34. B) 345. C) 3456. D) 34560
33) There are 15 points in a plane of which exactly 8 are collinear. The number of straight lines obtained by joining these points is ---- and the number of triangles formed with vertices at these points is ------
A) 78399 B) 7839 C) 673 D) 78
34) The total number of permutation of n(>1) different things taken not more than r at a time, when each thing may be repeated any number of times is
A) n! B) n! (n-1)/2
C) nʳ -1 D) n(nʳ-1)/(n-1)
35) 5 balls of different colours are to be placed in three boxes of different sizes. Each box can hold all five balls. The number of ways in which we can place the balls in the boxes so that no more remains empty is...
A) 148. B) 149. C) 150. D) 151
36) The number of ways in which an examiner can assign 30 marks to 8 Questions, giving not less than 2 marks to any question is..
A) ²¹C₅ B) ²¹C₆ C) ²¹C₇ D) none
37) The number of rectangles that you can find on a chessboard is..
A) 1286. B) 1296. C) 1200. D) n
38) The number of integers greater than 7000 that can be formed with the digits 3, 5,7,8 and 9. no digit being repeated, is
A) 162 B) 172 C) 182 D) 192
39) 10 persons amongst whom are A, B and C are to speak at a function. The number of ways in which it can be done if A want to speak before B, and B wants to speak before C is...
A) 6! B) 10! C) 10!/6 D) none
40) If x∈ ℕ and ˣ⁻¹C₄ - ˣ⁻¹C₃ -5/4. ˣ⁻²P₂ < 0, Then x is..
A) 3≤x≤10 B) 3,4,5
C) 6,7, 8,9 D) none
41) The largest integer n for which 35! divisible by 3ⁿ is...
A) 10. B) 12. C) 14. D) 15
42) If n is a positive integer, the value of
E= ²ⁿ⁺¹C₁ + ²ⁿ⁺¹C₂ + .... + ²ⁿ⁺¹Cₙ - ²ⁿ⁺¹C₂ₙ₊₁ ⁻²ⁿ⁺¹C₂ₙ - ... - ²ⁿ⁺¹Cₙ₊ is
A) 0 B) 1 C) -1 D) none
43) The number of natural numbers which are smaller than 2.10⁸ and which can be written by means of the digits 1 and 2 is.
A) 760. B) 762. C) 764. D) 766
44) The number of positive integral solution of x² - y² = 352706 is..
A) 0 B) 2 C) 3 D) 4
45) Let = {1,2,3, ...., n}. If X denotes the set of subset of A containing exactly three elements, then ∑min(P) P∈ X is
A) ⁿ⁺¹C₄ B) ⁿC₄. C) ⁿC₁₀ D) n
46) 10 person arranged in a row. The number of ways of selecting four persons so that no two persons sitting next to each other are selected is...
A) 39. B) 35. C) 30. D) 28
47) There are 15 seats in the First row of a cinema hall. The torch man has the instruction that the seat number 6 must be occupied. The number of ways in which 4 seats of the First row can be allotted so that no two of them are consecutive is....
A) 120. B) 124. C) 126. D) 128
48) The number of positive integral solutions of the inequality 3x + y +z≤30 is...
A) 120. B) 1200. C) 1210. D) 1215
49) if all permutations of the letters of the word RAKSHIT are arranged as in dictionary, then the five hundred and forty first word is....
A) RAKIT. B) RAKSHT
C) RAKHSIT D) RAKSHIT
50) The number of n-bit( digits 0 and 1) strings having exactly k is, with no 2 zeroes consecutive, is... (assume that 2k <n).
A) n B) n! C) nCr D) none
51) The value of 1.1!+ 2.2!+ 3.3! + .... + n.n! is....
A) n! B) (n+1)! C) (n+1)! -1 D) none
52) If (mn+1) pairs of letters are written down, each pair consisting of one chosen from the m letters, a₁, a₂, .... aₘ and the other from the letters b₁ b₂, ..., bₙ. then at least two pairs are identical.
A) True B) False
53) ²ⁿPₙ = 2ⁿ(1.3.5. ....(2n-1)).
A) true B) false
54) 2.6.10.....(4n-6)(4n-2)= (n+1)(n+2)...(2n-1)(2n).
A) true B) False
55) the greatest value of ²ⁿCᵣ (0 ≤r≤2n) is ²ⁿ Cₙ₋₁
56) The product of 2n consecutive negative integers is divisible by (2n)!
A) true. B) false
57) There cannot exist two positive integers n and r such that ⁿCᵣ, ⁿCᵣ₊₁, ⁿCᵣ₊₂ and ⁿCᵣ₊₃ and in AP.
A) true. B) false
58) If n(>1) and r are positive integer and ³ⁿCᵣ = ³ⁿCₙ₊ᵣ then each of them is equal to (3n)(3n-1)(3n-2) ..... (2n+1).
A) true. B) false
59) If n > 7, then ⁿ⁻¹C₃ + ⁿ⁻¹C₄> ⁿC₃
A) True. B) false
60) For n ≥ 4, 1!+2!+ ....n! cannot be the square of a positive integer
A) true B) false
61) given a positive integers n, there exist n consecutive positive integers, none of which is prime
A) true. B) false
62) ²ⁿCₙ < 4ⁿ/√(2n+1) for each positive integer n.
A) true B) false
63) If ⁿCᵣ₋₁ = 36, ⁿCᵣ= 84 and ⁿCᵣ₊₁ = 126, then ʳC₂ = 15
A) true. B) false
64) The number of ways in which 5 beads of different colours can form a necklace is 12.
A) true. B) false
65) The greatest value of ²ⁿCᵣ is double the greatest value of ²ⁿ⁻¹Cᵣ
A) True B) false
66) The Equation 1!+2!+ ..... x! = y³ has just one solution in the set of natural numbers.
A) true. B) false
67) From any set of 52 integers it is always possible to choose two integers such that their sum or difference is divisible by 100
A) true B) false
68) for x belongs to R, let [x] denotes the largest integer less than or equal to x. then value of
E= [1/3]+ [1/3+ 1/100]+[1/3+ 2/100]+ .....+ [1/3+ 99/100] is 33
A) true B) false
69) the number of non-negative Integral solutions of the Equation 2x+ y = 40 such that x≤ y is 20
A) true B) false
70) The number of integeral solutions of x₁ + x₂+ ...xₖ= n where n≥ k(k+1)/2 and x₁ ≥ 1, x₂≥2,...xₖ≥k is ⁿCₖ - n.
A) true. B) false
71) If a₁, a₂, ... a₁₀₁ is an arrangement of the number, 1,2, ...., 101, then the Product. (1- a₁)(2- a₂) ....(101 - a₁₀₁) is an even Integer.
A) true. B) false
72) there are 5 balls of different colours and 5 boxes of the same colours as those of the balls. The number of ways in which the balls one each in a box, could be placed so that no ball goes to the box of its own colour is 44.
A) true. B) false
73) The number of positive integers from 1 to 10⁶ (both inclusive) which are neither perfect squares, not perfect cubes, no perfect fourth powers is 978910.
A) true. B) false
74) There are 20 persons among whom are two brothers. The number of ways in which we can arrange them around a circle so that there is exactly one person between the two brother is (2)(18!).
A) true. B) false
75) If n is even, then 4ⁿC₀ < ⁿC₂ < ... ⁿCₙ/²
A) true B) false
75) A question paper is split into two parts--- partA and Part B. Part A contains 5 questions and part B has 4. Each question in partA has an alternative. A student has to attempt at least one question from each part. Find the number of ways in which the students can attempt the question paper.
A) 3600 B) 3620 C) 3630 D) none
76) A is a set containing n elements. A subset of P₁ of A is chosen. the set A is reconstructed by replacing the element of P₁. Next, a subset of P₂ of A is chosen and again the set is reconstructed by replacing the element of P₂. in this way subsets of a find the number of ways to m(>1) subset P₁,P₂,..... Pₘ of A are chosen. Find the number of ways of choosing P₁, P₂ , .....,Pₘ so that
i) P₁∩P₂∩......Pₘ= ¥
ii) P₁∪P₂∪.......∪Pₘ= A
A) (2ᵐ-1) B) 2ᵐ C) (2ᵐ-1)ⁿ D) non
77) What can be the maximum population of a country in which no two persons have an identical set of teeth. (disregard of shape and size of the teeth. take only the positioning of the teeth in consideration). Also, assume that there is no person without a tooth and no person has more than 32 teeths)
A) 2³². B) 2³² -1. C) n³². D) n
78) how many different car licence plates can be constructed if the licences contain three letters of the English alphabet followed by a three digit number if repetition are allowed ? And if repetation are not allowed.
26³. 999, 26P3. 10P3
79) If X is a set containing n elements and Y is a set containing m elements, how many functions are there from X to Y? How many of this functions are one to one. mPn
80) Prove that (n!)! is divisible by (n!)^(n-1)!
81) m men and n women are to be seated in a row so that no two women sit together. If m> n. show that the number of ways in which they can be seated is {m!(m+1)!}/(m-n+1)!
82) A tea party is arranged for 2m people along two sides of a long table with m chairs on each side, r men wish to sit on one particular side and s on the other. In how many ways can they be seated ? (Assume that r, s ≤m)
ᵐPᵣ. ᵐPₛ. ²ᵐ⁻ʳ⁻ˢP₂ₘ₋ᵣ₋ₛ
83) A family consists of a grandfather, m sons daughters and 2n grantedchilren. they are divine to be seated row for dinner. The grandchildren wish to occupy the n seats at each end and the grandfather refuses to have a grandchild on either side of him. In how many ways can the family be made to sit?
(2n)! m! (m-2)!
84) Find the number of ways of arranging 21 white balls and 3 black balls in a row so that no two black balls are placed together. 1540
85) A man has 7 relatives, 4 women and 3men. His wife also 7 relatives, 3 women and 4men. In how many ways can they invite 3 women and 3 men so that 3 of them are the man's relatives and 3 his wife's? 485
86)a) In how many ways can a pack of 52 cards be divided equally among 4 players in order? 52!/(13!)⁴
b) in how many ways can you divide these cards into 4 sets, three of them having 17 cards each and the fourth just one card? 52!/(17!)³.3!
87) support a city has m parallel roads running east- west and n parallel roads running North- South. How many Rectangles are formed with their sides along these roads? If the distance between every consecutive pair of parallel roads is the same. how many shortest possible routes are there to go form one corner of the city to its diagonally opposite corner? {mn(m-1)(n-1)}/4
88) A box contains 2 white, 3 black and 4 red balls . in how many ways can three balls are drawn from the box if at least one black ball is to be included in the draw. 64
90) there were two women participating n a chess tournament. Every participant played two games with every other participants. the number of ggmes that men played between themselves proved to exceeds by 66. compared to the number of games the men played with women. how many participants were there ? how many games were played? 156
91) Mr. A has x children by his first wife Ms. B has x+1 children by her first husband. they marry and have children of their own. The whole family has 10 children. assuming that two children of the same parents do not fight. find the maximum number of fights that can take place among children. 3
92) Bhawna has 4 different toys and Quincy has 7 different toys. find the number of ways in which they can exchange their toys so that each keeps her initial number of toys. 329
93) A train going from Delhi to Jaipur stops at 7 intermediate stations. five person enter the train during the journey with five different tickets of the same class. how many different set of tickets they could have had? 28
94) 6 apples and six oranges are to be distributed among 10 boys so that each boy receives at least one fruit. Find the number of ways in which the fruits can be distributed. 26250
95) Find the number of ways in which 16 apples can be distributed among four persons so that each of them gets at least one Apple. 455
96) find the number of whole numbers formed on the screen of a calculator which can be recognised as numbers with (unique) Correct digits when they are inverted. The greatest number that can be formed on the screen of the catalogue is 999999. 100843
97) Find the number of natural numbers which are smaller than 2. 10⁸ and are Divisible by 3 and which can be written by means of the digits 0,1 and 2. 4373
98) suppose there are six points in a plane no three of which are collinear. each point is connected to other by line segment and this line segment is coloured either red or blue. show that irrespective of the ways we colour the connecting line segments, there always exists a triangle with vertices at these points and whose sides are of the same colour.
99) If 101 Integers are selected from the set S={1,2,....200} prove that among the selected there exist two distinct integers such that one of them is a multiple of the other. r
100) At a party there are more boys than girls. If each boy dances with exactly 2 girls, prove that there is at least one girl who dances with at least 3 boys.
101) Find the number of ways of putting 5 distinct rings on 4 fingers at the left hand.(Ignore the difference in the size of rings and the fingers). 6720
102) find the number of ways arranging r district objects in distinct boxes. (n+r-1)!/(n-1)!
103) find the number of ways of placing r identical objects in to n distinct boxes. (n+r-1)Cr
104) In any set of ten 2 digit numbers show that there always exist two non empty disjoint subsets A and B such that sum of the numbers in A is equal to sum of the numbers in B.
105) if N distinct things are arranged in a circle, show that the number of ways of selecting three of these things so that no two of them are next to each other.
106) show that there cannot exist two positive integers n and r for which ⁿCᵣ,ⁿCᵣ₊₁,ⁿCᵣ₊₂ are in H. P
107) The number of ways in which 7 distinct toys can be distributed among 3 children is..
A) 3⁷ B) 7³ C) 35. D) 35
108) Each of the five Questions in a multiple choice test has 4 possible answers. The number of different sets of possible answer is..
A) 4⁵-4. B) 5⁴-5. C) 1024 D) 624
109) A set contains (2n+1) elements. The number of subsets of the set which contains more than n elements is..
A) 2ⁿ⁻¹. B) 2ⁿ. C) 2ⁿ⁺¹ D) none
110) A is a set containing n elements. A subset P of A is chosen. The set A is reconstructed by replacing the elements of P. A subset Q of A is again chosen. The number of ways of choosing P∩Q contains exactly two elements is
A) 9. ⁿC₂ B) 3ⁿ- ⁿC₂ C)2.ⁿCₙ D) non
111) 10 different letters of an alphabet are given. words with 5 letters are formed from these given letters. The number of words which have at least one of their letters repeated is
A) 69760. B) 30240
C) 99748. D) none
112) The number of ways in which we can select four numbers from 1 to 30 so as to exclude every selection of four consecutive numbers is..
A) 27378. B) 27405
C) 27397. D) none
113) In ⁿC₄, ⁿC₅ and ⁿC₆ are in A. P ., The value of n can be..
A) 14 B) 11 C) 7 D) 8
114) The letters of the word SURITI are written in all possible orders and these words are written out as in a dictionary. then the rank of the word SURITI is
A) 236. B) 245. C) 307. D) 315
115) A candidate is required to answer 6 out of 10 questions which are divided into two groups, each containing 5 Questions. He is not permitted to attempt more than 4 questions from either group. The number of different ways in which the candidates can choose 6 question is.
A) 50. B) 150. C) 200. D) 250
116) there are n different books and p copies of each in a library. The number of ways in which one or more than one book can be selected is..
A) pⁿ+1. B) (p+1)ⁿ - 1
C) (p+1)ⁿ- p. D) pⁿ
117) If K= ᵐC₂ where m ≥ 2 then the value of ᵏC₂ is given by
A) ᵐ⁺¹C₄ B) ᵐ⁻¹C₄.C) ᵐ⁺²C₄ D) n
118) There are n white and n black balls marked 1,2,..., n. The number in which we can arrange these balls in a row so that neighbouring balls are of different colours is...
A) n! B) (2n)!
C) 2(n!)² D)(2n)!/(n!)²
119) The number of squares which we can form on a chess board is
A) 64. B) 160. C) 224. D) 204
120) The number of ways in which we can arrange four letters of the word MATHEMATICS is given by
A) 136. B)2454. C)1680. D)192
121) The number of ways in which we can distribute mn students equally among m sections is given by
A)(mn)!/n! B) (mn)!/(n!)ᵐ
C) (mn)!/m!n! D)(mn) ᵐ
122) If a polygon has 54 diagonals, the number of its sides is given by given by
A) 12. B) 11. C) 10. D) 9
123) out of 10 white, 9 black and 7 red balls, the number of ways in which one or more balls can be selected if given by.
A) 881. B) 891. C) 879. D) 892
124) The number of ways of dividing 2n people into n couples is__
A)(2n)! B) (2n)!/2ⁿ
C) (2n)!/2ⁿ(n!). D) none
125) The number of ways of arranging 2m white and 2n red counters in a straight line so that each arrangement is symmetrical with respect to a Central mark is ___
A) (m+n)!/m!.n! B) (m+n)!
C) m!. n! D) none
126) The number of ways in which ten candidates A₁, A ₂,....A ₁₀ can be ranked so thatA₁ is always above A is____
A) 9!/2. B) 9! C) 10! D) 10!/2
127) The number of ways of arranging p positive and n(≤ p+1) negative signs in a row so that no two negative signs are together is _____
A) ᵖ⁺¹Cₙ B) ᵖPₙ D)ᵖCₙ -1 D) none
128) On Diwali, all the students of a class send greeting cards to one another. if the postman delivers 1640 greeting cards to the students of this class, then the number of students in the class is______
A) 40. B) 41 C) 42 D) 43
129) If m point on one straight line are joined to n points on another straight line, then excluding the given points, the two lines will intersect at_____
A)mn. B) mn/2 (m-1)(n-1)
C) m C n. D) none
130) If a ,b ,c are (m+1) distinct prime numbers, then the number of factors of aⁿbc... is_____
A) 2ᵐ B) 2ᵐ -1C) 2ᵐ(n+1) D) none
131) let A be a set of n distinct elements. then the total number of distinct functions from A to A is _____ and out of these____ these are onto function.
A) nⁿ. B) n. nⁿ. C) nⁿ. n! D) none
132) A box contains 2 white 3 black and 4 red balls. The number of ways in which we can select balls from the box, if at least one black ball is to be included in the selection, is _____
A) 60. B) 62. C) 63. D) 64
133) The side AB, BC and CA of a triangle ABC have 3,4 and 5 interior points respectively on them. The number Triangle that can be construed using these interior points as vertices is____
A) 292. B) 205. C) 305. D) 392
134) The value of the expression ⁿC₀ + ⁿ⁺¹C₁+ ⁿ⁺²C₂+...+ ⁿ⁺ʳCᵣ is____
A) n! B) (n+r+1)! C) ⁿ⁺ʳ⁺¹Cᵣ D) no
135) The number of 30 digit ternary sequences(sequences using only the digit 0 ,1 and 2) having exactly ten 1s is____
A) 2²⁰. B) 20². C) 2²⁰- 1. D) none
136) The number of integral solutions of x₁ + x₂ +x₃= 0 with x≥ - 5 is_____
A) 130. B) 134. C) 136. D) none
137) If m parallel lines are intersected by n parallel lines, number of parallelograms thus formed is _____
138) The result of 21 football matches (win, lose or draw) are to be predicted. The number of different forecasts which contain exactly contain 18 correct results is_____
139) A man invites a party of (m+n) friends to dinner and place m at one round table and n at another. The number of ways of arranging the guests is____
TRUE/ FALSE
140) If a, b, c,... k are positive integers such that a+b+c+...+k≤n, then n!/(a!. b!.....k!) is a positive integer.
141) if n= ab(a>1, b>1), then (n-1)! is divisible both a and b.
142) if a and b are two positive integers, then (ab)!/(a! b!ᵃ)) is an integer.
143) If n is any positive integer, then (n²)/(n!)ⁿ⁺¹ is an integer.
144) 7/¹⁰⁰⁰C₅₀₀
145) The number of 7 digit numbers, such that sum of their digits is odd is 45 x 10⁵
146) out of 10 consonant and four Vowels, the number of words that can be formed using 6 consonants and 3 vowels is ¹⁰P₆ x ⁶P₃ .
147) A question paper contains 6 questions, each having an alternative. The number of ways in which an examine can attempt one or more questions is 31.
148) The number of ways in which we can select four letters out of the letters of the word EXAMINATION is 136
149) n(≥3) distinct papers are set in an examination. of these exactly two are on mathematics. The number of ways of arranging the papers so that the two on mathematics are together is 2(n-1)!
150) The number of zeros at the end of 100! is 24.
151) There are n distinct white and n distinct black balls marked 1,2, ..,n. The number of ways in which we can arrange them in a row so that neighbouring balls are of different colours is ²ⁿCₙ .
152) The number of non congruent rectangles that can be formed on a chess board is 36.
153) If one quarter of all three subsets of the integers 1,2,.., m contains the integers 5, then m=10
154) The number of ways in which we can choose a committee from four men and six women so that the committee includes atleast two men and atleast twice as many women as men is 136.
155) For all positive integers n. ²ⁿCₙ + ²ⁿCₙ₋₁ = 1/2 (²ⁿ⁺²Cₙ₊₁).
156) The number of non negative integral solution of x₁ + x₂+x₃+x₄= 20 is 536.
157) Find the number of ways in which can put n distinct objects into two identical boxes so that no box remains empty.
A) 2ⁿ B) 2ⁿ⁻¹ C) 2ⁿ⁻¹- 1. D) none
158) If X is a set containing n(>1) elements and Y is a set containing m(>1) element, how many relations are there from X to Y ?
A) 2ᵐ B) 2ⁿ C) 2ᵐⁿ D) none
159) How many numbers each lying strictly between 100 and 1000 can be formed using the digit 1,2, 3 ,5 ,7 and 0 when the repetition of digits is allowed ?
A) 176 B) 177 C) 178 D) 179
160) Show that (kn)! is divisible by (n!)ᵏ
161) 10 guests are to be seated in a row. Three of them are to be seated together. of the remaining 7 do not wish to sit by side. Find the number of ways in which this can be done.
A) 180000. B) 181000
C) 181400. D) 181440
162) How many three digit numbers are of the form xyz with x, z< y, x≠ 0 ?
A) 200 B) 210 C) 220 D) 240
163) A valid FORTRAN identifier consists of a string of one to six alphanumeric characters beginning with a letter. How many valid FORTRAN identifiers are there ?
A) 26(36⁶-1) B) 36⁶
C) (36⁶-1)/35 D)26(36⁶-1)/35
164) an unlimited number of coupons bearing the letters A ,B and C are available. Find the number of ways of choosing m of these coupons so that they cannot be used to spell BAC.
A) 2ᵐ. B) 3. 2ᵐ.
C)(2ᵐ-1). D) 3(2ᵐ-1)
165) m numbers are chosen with replacement from the numbers 1,2,3, ..., n. Find the number of ways of choosing the numbers so that the maximum number chosen is exactly r(≤n).
A) rᵐ. B) (r-1)ᵐ
C) rᵐ(rᵐ-1). D) rᵐ-(r -1)ᵐ
166) Show that there cannot exist two positive integers n and r for which ⁿCᵣ , ⁿCᵣ₊₁ and ⁿCᵣ₊₂ are in GP.
167) How many n-digit numbers are there in which only the digit 1,2,3,.....,9 are used and in which no two consecutive digits are the same?
A) 9.8ⁿ. B) 8ⁿ⁻¹ C) 9. 8ⁿ⁻¹ D) none
168) A conference attended by 200 delegates is held in hall. The hall has 7 doors, marked A, B, ....G. At each door, an entry book is kept and the delegates entering through that door sign it in the order in a which they enter. If each delegates is free to enter any time and any through any door he likes, how many different sets of 7 lists would arise in all ? (Assume that every person signs only at his first entry)
A) ²⁰⁷P₂₀₀ B)²⁰⁶C₂₀₀ C) ²⁰⁸C₂₀₀ D) n
169) A straight is a 5 card hand containing consecutive values. How many different straight are there ? If, in addition, the cards are not all from the same suit, then how many straights are there ?
A) 9216, 9180 B) 9100, 9100
C) 9210, 8670 D) none
170) How many ways are there to spell RACHITHCAR by going from one letter to the adjacent one is the Fig. Below
R
A A
C C C
H H H H
I I I I I
T T T T T T
I I I I I
H H H H
C C C
A A
R
A) 250. B) 252. C) 254. D) 256
171) How many ternary sequences of length 9 are there which either begin with 210 or end with 210 ?
A) 1430 B)1420 C) 1431 D) 1421
172) Suppose that there are piles of red, blue and green balls and that each pile contains at least 8 balls
a) in how many ways can 8 balls be selected ?
A) 41 B) 42 C) 43 D) 45
b) in how many ways can 8 balls be selected if at least one ball of each colour is to be selected ?
A) 20. B) 21. C) 22. D) 23
173) prove (without using the Binomial Theorem) that
A) ⁿₖ₌₀∑ ⁿCₖ= 2ⁿ
B)ⁿₖ₌₀∑ (ⁿC ₖ)²= ²ⁿCₙ
C) ⁿₖ₌₀∑ 2ᵏ .ⁿCₖ = 3ⁿ
D) ⁿₖ₌₀∑ k.ⁿCₖ = n. 2ⁿ⁻¹
174) in how many ways can 15 identical mathematics books be distributed among 6 students?
A) 15000. B) 15500
C) 15504. D) none
175) How many integral solutions are there to x+y+z+t= 29, when x> 0 , y>1, z>2 and t≥0?
A)2000. B) 2600 C) 2602. D) none
176) how many integral solutions are there to the system of equations x₁+x₂+x₃+x₄+x₅= 20 and x₁+x₂+x₃= 5 when xₖ≥0?
A) 300. B) 330. C) 333. D) none
177) From the number of positive, unequal integeral solutions of the Equation x+y+z+t= 20.
A) 2ⁿ⁻¹ B) 2ⁿ⁻¹(n²+8)
C) 2ⁿ⁻³(n²+8) D) 2ⁿ⁻³(n²+7n+8)
178) Show that the number of ways of selecting n objects out of 3n objects, of which n are alike and the rest are different, is
2²ⁿ⁻¹+ (2n-1)!/{n!(n-1)!}
179) An examination consists of four papers. each paper has a maximum of m marks. Show that the number of ways in a which a student can get 2m marks in the examination is 1/3{(m+1)(2m²+4m+3)}.
180) m equip spaced Horizontal lines are intersected by n equal spaced vertical lines. if m < n and the distance between two successive horizontal lines is the same as the distance between two successive vertical lines, show that the number of squares formed by these lines is 1/6{m(m-1)(3n-m-1)}.
181) if m and n are positive integers such that m≤ n and m+n = a, where a is a fixed positive integer, show that the greatest value of m(m-1(3n-m-1) is 1/4 a(a-1)(a-2) or 1/4 (a+1)(a-1)(a-3), according as a even or odd.
182) If a straight rods of indefinite length are placed so as to form the greatest possible number of squares, prove that this number is 1/24 a(a-1)(a-2) or 1/24 (a+1)(a-1)(a-3) according as a is even or odd.
183) The sides of a triangle a,b and c, where a, b and c are integers and a≤ b≤ c. If c is given, show that the number of different Triangle is 1/4 c(c+2) or 1/4 (c+1)², according as c is even or odd. Also, show that the number of isosceles or Equilateral Triangle is 1/2 (3c-2) oor 1/2 (3c-1), according as c is even or odd.
184) Each side of a triangle is an integral number in cm, no side exceeding c cm. Prove that the number different triangles which can be so formed is 1/24 c(c+2)(2c+5) or 1/24 (c+1)(c+3)(2c+1), according as c is even or odd. Also, show that the number of isosceles or equilateral triangles is 3c²/4 or (3c²+1)/4, according as c is even or odd.
185) If out of 3n letters there are n As, n Bs and n Cs, show that the number of ways of selecting r letters out of these is the same as the number of ways of selecting 3n - r letters out of them. If n< r < 2n+1, Show that the number of ways of selecting r letters is given by 1/2 (n+1)(n+2) + (r-n)(2n-r) and that the maximum number of such selections is 1/4 {3(n+1)²+1} or 3/4 (n+1)² according as n is even or odd.
186) 18 guests have to be seated, half on each side of a long table. Four particular guests wish to sit on one particular side of the table and another 3 on the other side. determine the number of ways in which the seating arrangements can be made.
A) ¹¹C₅. B) ¹¹C₅. 8!
C) ¹¹C₅. 9! D) ¹¹C₅. 9!. 9!
1) The number of different algebraic expressions that can be made by combining the latters p, q, r, s and t in this order with the signs '+' and '-', taking all the letters together is..
A) 21. B) 23. C) 31. D) 32
2) th8e number of different factors of 2160 is..
A) 29. B) 39. C) 49. D) none
3) 8 different chocolates can be distributed equally between two boys in ..
A) 70 B) 35 C) 38 D) 19
4) From a group of persons the number of ways of selecting 5 persons is equal to that of 8 persons. The number of persons in the group is...
A) 29 B) 25 C) 13 D) 11
5) A man has 5 oranges and 4 mangoes. How many different selections having at least one orange is possible?
A) 25 B) 30 C) 35 D) 40
6) A man has 6 friends. The number of ways in which he can invite one or more of them to his house is..
A) 6! B) 6!- 1 C) 2^6! D) none
7) A 5-digited number is divisible by 3 and it is is formed by using 0 ,1 2, 3 , 4 and 5 5 without repetation. The total number of ways in which such a number can be formed is..
A) 126. B) 216. C) 621. D) 261
8) All the letters of the string 'AEPRAB' are arranged in all possible ways. The number of such arrangements in which two Vowels are not adjacent to each other is..
A) 220 B) 115 C) 72 D) 65
9) The number of ways in which the letter of the string the string ANRTIPF can be arranged so that the vowels may appear in the odd places is..
A) 1230 B)1350 C)1440 D)1570
10) if there are 10 person in a gathering gathering in a gathering in a gathering in a gathering and if each of them shakes hand with everyone else. then the number of handshakes that takes place in the gathering is...
A) 20. B) 45. C) 2¹⁰. D) 10²
11) The number of parallelograms that can be formed from a set set of 4 parallel lines intercecting another set of three parallel lines is ..
A) 21 B) 20 C) 18 D) 16
12) The number of students to be selected at a time from a group of 16 students, so that the number of selection is the greatest is...
A) 16 B) 14. C) 8. D) none
13) The number of different arrangements with the letters of the word ALGEBRA so that the two A's are not together is..
A)1800. B)2520. C)720. D) none
14) The number odd integers of Six significant digits that can be formed with the digit can be formed with the digit be formed with the digit with the digit 0,1,4,6,7 without repetition of the digit is..
A) 96. B) 108. C) 266. D) 288
15) The number of words that can be made by writing down the letters of the word CALCULATE such that each word starts and ends with consonant is..
A) 7! B) 7!/2 C) 5. 7!/2 D) 9. 7!/2
16) The number of triangles that can be formed with 10 points as vertices, k of them being collinear, is 110. Then the value of k is..
A) 3. B) 5. C) 7. D) none
17) The number of ways in which 5 '+' sign and 3 'X' sign can be arranged in a row is..
A) 56. B) 65. C) 72. D) 81
18) The number of ways in which 15 Class XI students and 12 class XII students be arranged in a line so that no two class XII students may occupy consecutive positions is..
A) 12!.16!/4! B) 15!.13!/4!
C) 16!.13!/4! D) 15!.16!/4!
19) the number of strings of three letters that can be formed with the letters choosen from CALCUTTA is.
A) 48. B) 62. C) 96. D) 102
20) The number of permutation of the letters of the word MADHUBANI where the arrangements do not begin with M but end with I is..
A) 16740. B) 17460
C) 14670. D) none
21) the number of ways in which A committee of 5 persons may be formed out of 6 Men and 4 women under the condition that at least one woman has to be selected necessarily is..
A) 252. B)246. C) 242. D) none
22) given that balls of the same colour are identical, the number of ways in which 18 white balls and 19 a red balls may be arranged in a row so that no two white balls may come together is
A) 180. B) 190. C) 200. D) 210
23) in an examination there are three multiple choice questions and each question has four choices. The number of a ways in which one fail to get all answers correct is..
A) 12 B) 21 C) 36 D) 63
24) The number of diagonals that can be drawn by joining the vertices of an octagon is
A) 28. B) 20. C) 18. D) 16
25) out of 6 given point 3 are collinear. The number of different straight lines that can be drawn by joining any two points from those 6 given points is..
A) 12. B) 10. C) 9. D) none
26) The total number of selections of at least one red ball from 4 red balls and 3 Blue balls, if the balls of the same colour are different,is
A) 95. B) 105. C) 120. D) 125
27) in an examination of 9 papers a candidate has to pass in more papers than the number of papers in which he fails in order to be successful. The number of ways in which he can be unsuccessful is..
A) 265. B) 255. C) 256. D) 625
28) The number of integers greater than 50000 that can be formed by using the digits 3,5, 6,6,7 is..
A) 54 B) 46 C) 32 D) none
29) The number of arrangement that can be formed from the letters of the word VIOLENT, so that the vowels may occupy only odd position is..
A)576. B)574. C) 572. D) none
30) in a group of 15 boys there are seven Boy Scouts. The number of ways in which 12 boys can be selected from the group so as to include atleast six Boy Scouts is...
A) 125. B) 127. C)52. D) 255
31) 15 distinct objects may be divided into three groups of 4,5 and 6 objects in
A) 230230. B) 320320
C) 360360. D) 630630
32) The number of different ways in which 1440 can be expressed as the product of two factors is..
A) 18. B) 16. C) 14. D) none
33) The number of different rectangles (regarding every square as a rectangle as well) that are there on a chess board is
A) 1280 B)1284 C) 1296 D)1300
34) The number of arrangements which can be made out of the letters of the word ALGEBRA without changing the relative positions of the Vowels and consonants is..
A) 54. B) 64. C) 70. D) 72
35) The number of factors of 420 is..
A) 22. B) 23. C) 24. D) none
36) A boat has a crew of 10 men of which 3 can row only on one side and two only on the other. The number of ways the crew can be arranged in the boat is..
A) 142000. B) 144000
C) 124000. D) none
37) There are 10 points in a plane of which no 3 Points are collinear and 4 points are concyclic. The number of different circles that can be drawn through at least three points of these points is..
A) 117. B) 120. C) 122. D) 124
38) The number of 6 digited integers that can be made using the digits 3 and 4 and in which at least two digits are different, is..
A) 60. B) 61. C) 62. D) none
39) The sum of digits in unit place of all the four digited numbers formed with the help of 2,3,4,5, taken all at a time is..
A) 54. B) 108. C) 84. D) none
40) The number of different ways in which 15 distinct object may be divided into three groups of five objects each is..
A) 216216. B) 126126
C) 216612. D) 126612
41) The number of different arrangements that can be made out of the letters of the word ALLAHABAD, such that the Vowels may occupy the even position only is..
A) 70. B) 50. C) 60. D) 120
42) The number of ways in which 4 letters can be posted in 3 post boxes is..
A) 256 B) 81 c) 12 D) none
43) at an election, a voter may vote for any number of candidates, not greater than the number to be elected. There 10 candidates and 4 are to be elected. If a voter votes for at least one candidate, then the number of ways in which he can vote is.
A) 385 B) 1110 C)5040. D) 6210
44) How many ways are there to arrange the letters in the word GARDEN with the Vowels in alphabetical order ?
A) 360 B) 240 C) 120 D) 480
45) The number of a ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is...
A) 38. B) 21. C) 5. D) 56
46) The number of different solutions (x,y,z) of the equation x+y+z= 10, where each of x,y,z is a +ve integer is..
A) 36. B) 75. C) 990. D) none
47) 5. ⁿP₄= 6. ⁿ⁻¹P₄ then n is...
A) 22 B) 24 C) 48 D) 50
48) ⁿ⁺¹P₄ : ⁿ⁻¹P₃ = 72:5 then n is..
A) 8 B) 9 C) 10 D) 11
49) ¹⁰Pᵣ = 5040 then r is..
A) 2 B) 3 C) 4 D) 5
50) ⁿPᵣ = ⁿPᵣ₊₁ and ⁿCᵣ = ⁿCᵣ₋₁ then values of n and r ....
A) 2,5 B) 3,2 C) 6,3 D) 7,4
51) ¹²Pᵣ= ¹¹P₆ + ¹¹P₅ Then r is
A) 3. B) 4. C) 5. D) 6
52) If (n+2)!= 2550. n! Then n is..
A) 47. B) 48. C) 49. D) 50
53) ¹⁰Pᵣ= ⁹P₅ + ⁹P₄ then r is
A) 2. B) 3. C) 4. D) 5
54) ⁴⁻ˣP₂ = 6 then x is...
A) 1 B) 2 C) 3 D) none
55) ²ⁿ⁺¹Pₙ₋₁ : ²ⁿ⁻¹Pₙ= 3:5 then n is.
A) 4. B) 3. C) 2. D) none
56) ⁿCᵣ₊₁ + ⁿCᵣ₋₁ + 2.ⁿCᵣ is =
A) ⁿ⁺²Cᵣ B) ⁿ⁺²Cᵣ₊₁
C)ⁿ⁺¹Cᵣ. D) ⁿ⁺¹Cᵣ₊₁
57) ¹⁶Cᵣ = ¹⁶Cᵣ₊₂ Then ʳC₄ is equal
A) 21. B) 27. C) 35. D) 39
58) IF 3.ˣ⁺¹C₂+ ²P₂. x = 4.ˣP₂ Then x is...
A) 2 B) 3 C) 5 D) 7
59) If ²ⁿC₃ : ⁿC₂= 44:3 then n is..
A) 3. B) 4. C) 5. D) 6
60) if ⁿP₅ = 60.ⁿ⁻¹P₃ Then n is..
A) 8 B) 10 C) 12 D) 14
61) The value of ²⁰C₅ + ⁵ⱼ₌₂∑²⁵⁻ʲC₄ is...
A) 24504. B) 44502
C) 42504. D) 45042
62) If ⁿC₄= 21. ⁿ⁾²C₃ then n is
A) 6. B) 10. C) 12. D) 14
63) The value of ⁴⁰C₃₁ + ¹⁰ⱼ₌₀∑⁴⁰⁺ʲC₁₀₊ⱼ is equal to..
A) ⁷²C₃₁ B) ²⁷C₈ C) ¹⁹C₁₁. D) ⁵¹C₂₀
64) ²⁸C ₂ᵣ : ²⁴C₂ᵣ₋₄ = 225: 11 then value of r is..
A) 7. B) 9. C) 14. D) none
65) IF ⁵⁶ᵣ₊₆ : ⁵⁴Cᵣ₊₃ = 30800 :1, then the value of r is..
A) 11 B) 21 C) 31 D) 41
66) If ⁿ⁻¹C₃+ ⁿ⁻¹₄a> ⁿC₃, then
A) n<. B) n>7. C) n<5. D) n>6
67) ⁿ⁺²C₈ : ⁿ⁻²C₄ = 171:2 then n is.
A) 18 B) 19 C) 20 D) 21
68) ⁴⁷C₄ + ⁵ᵣ₌₁∑⁵²⁻ʳC₃ is
A) 277025 B) 275027
C) 507227 D) 270725
69) ⁿPᵣ= 504 and ⁿCᵣ= 84, then n .
A) 3. B) 6. C) 9. D) 12
70) If ²ⁿC₁ +²ⁿC₂ +.....+²ⁿCₙ₋₁ + 1/2 ²ⁿCₙ = 127, then n is equal..
A) 4. B) 5. C) 3. D) none
71) The maximum number of point into which 4 circles and 4 straight lines is:
A) 26. B) 56. C) 50. D) 72
72) Number of triangles formed by 12 points, of which 7 are in the same in the same the same straight line is:
A) 185 B) 220. C) 92. . D) None
73) (ⁿPᵣ₋₁)/a = (ⁿPᵣ)/b=(ⁿPᵣ₊₁)/c, then b²/{a(b+c)}
a) 1 b) 2 c)1/2 d) none
74) How many four digit numbers can be formed using the numbers 0,1,2,3,4,5,6,7,8,9.
a) 5040 b) 504 c) 4536 d) 5544
75) ⁿ⁺⁵Pₙ₊₁= 11(n-1)/2 . ³⁺ⁿPₙ, then the value of k is
a) 6,7 b) 1,2. c) 3,4. d) 5,6
76) How many words can be formed with the letters of the word ORDINATE so that the vowels will be in odd places.
a) 576 b)676 c) 276 d) 625
77) The value of ¹⁴C₁+¹⁴C₃ + ¹⁴C₅ + ...¹⁴C₁₃ is.
a) 2¹⁴ - 2 b)2¹³. c)2¹⁰. d) 2¹¹
78) In how many ways 20 same category category Apples can be divided amongst 4 persons
a) 1771 b) 1700 c) 1770 d) 1991
79) Number of straight lines lines that can be formed by joining 20 points of which 4 points are collinear are are collinear are points are collinear are are collinear are
a) 160 b) 185. c) 190. d) none
80) ⁿ⁻¹C₁ + ⁿ⁻¹C₄ > ⁿC₃ if n is greater than---
a) 5 b) 6 c) 4 d) 7
81) Number of ways of arranging 5 boys and three girls girls in a row so that no two girls are together is--
a) 5!3! b)⁶C₃5!3! c) 25!3! d) none
82) In how many ways seven digited numbers can be selected so that sum of the digit would be even.
a) 45.10⁵ b) 45.10⁶
c)47.10⁵ d) 47.10⁶
83) In a college of 300 students, every students read 5 newspaper newspaper and every newspaper read by 60 students students. The number of newspaper is..
a) atleast 30 b) almost 20
c) exactly 25 d) none
84) From 4 officers and 8 Jawans in how many way can be 6 chosen to include at least one officer..
a) 896 b) 986 c) 1096 d) none
85) The sum of the digits in the units place of all place of all all numbers formed with the the help of 3,4,5,6 taken all at a time..
a) 18 b) 108 c) 432. d) 144
86) Six identical coins are arranged in a row. The number of ways in which the number of tails is equal to the number of head is is..
a) 20 b) 120 c) 9 d) 40
87) ⁿPᵣ = ⁿPᵣ₊₁ and ⁿCᵣ = ⁿCᵣ₋₁, then (n, r) is...
a) 2, 3 b) 3,2 c) 4,2 d) 4,3
88) The number of ways in which a mixed doubles doubles tennis game can be arranged between 10 between 10 players consisting of 6 men and 4 women is...
a) 180 b) 90 c) 48 d) 12
89) If there are 12 persons in a party party and if each two of them of them shake hands with each other, how many handshakes happen in the party.
a) 66 b) 69 c) 72 d) 75
90) The number of combination of n different things when at least one object is taken at a time is..
a) 2ⁿ b) 2ⁿ -1 c) n(n+1)/2 d) N
91) There are 13 stations on a certain certain railway line. How many kinds of different single third class ticket class ticket have to be printed in order that it may be possible to travel from any station any other station..
a) 144 b) 156 c) 169 d) None
92) How many numbers can be made with the digits 3,4,5,6,5,4 and 3 so that odd digits may occupy odd places...
a) 18 b) 16 c) 21 d) 24
93) Number of ways of placing placing 5 different balls in 3 different boxes (No box given empty)..
a) 10 b) 15 c) 25 d) 150
94) The number of rectangle which can be formed by on a chess board is..
a) 81 b) 1296 c) 646 d) 324
95) The expressionⁿ⁺¹Cᵣ +3 ⁿ⁺¹Cᵣ₋₁ . 3 ⁿ⁺¹Cᵣ₋₂ + ⁿ⁺¹Cᵣ₊₃ is equal to --
a) ⁿ⁺³Cᵣ b)ⁿ⁺⁴Cᵣ c)ⁿ⁺⁵Cᵣ d) None
96) There are 15 bulbs in a room. Each of them can be operated independently. The number of ways in which the room can be lighted.
a) 8⁵ - 1 b)(32)² -1
c)(32)³ - 1 d) 8⁴ - 1
97) In how many ways can the letter of the word SUCCESS be rearranged so that 2 C are together, but no two S are together
a) 20 b) 24 c) 28 d) 32
98) A polygon has 44 diagonals, then the number of its sides are..
a) 11 b) 7 c) 8 d) None
99) ²⁴Cᵣ = ²⁴C₂ᵣ₋₁ ,Find r..
a) 8 b) 7 c) 6 d) 9
100) A code word consisting of two distinct English alphabets followed by two distinct numbers from 1 to 9 9 For example CA 23 is a code word code word word. The total number of code word is..
a) 46800 b)20800
c)46000 d)46300
101) ⁿP₄ = 12 ⁿP₂ , then find n is..
a) 1 b) 6 c) 2 d) 8
102) ¹⁰Pᵣ = 5040, the value of r is..
a) 2 b) 4 c) 6 d) 3
103) Keeping all the vowels together, how many words can be formed using the letters of the word MECHNIC ..
a) 24 b) 2160 c) 720 d) 120
104) The sum of all the four digit digit numbers formed by the digits 3,4,5,6 is..
a) 5040 b) 119988 c) 911988 d) N
105) find out the number of words generated from the word ARRANGE in such a way that no two R will be together.
a) 900 b) 700 c) 1620 d) N
106) how many different signals can be given given with the aid of 5 flags of different colours ..
a) 24 b) 120 c) 325 d) 75
107) find out the number of 4 digits formed by the digit the digit 1,2,3,4,5..
a) 120 b) 54 c) 124 d) 71
108) how many words can be made, keeping serially 3 vowels in the middle, from the letters of the word INTERIM....
a) 120 b) 24 c) 72 d) 5040
109) Number of corners of the 12-sided polygon is...
a) 66 b) 12 c) 20 d) 54
110) The value if ⁿPᵣ and ⁿCᵣ , becomes equal when the value of r is...
a) 0 b) 1
c) a and b d) a and b wrong
111) If ²ⁿC₃ is to ⁿC₃= 11 is to 1 then the value of n is ...
a) 6 b) 8 c) 3 d) 7
112) Value of ²⁰C₅ + ⁵ⱼ₌₂∑²⁵⁻ʲC₄ is..
a) 4200 b) 4002 c) 42504 d) 3020
113) ⁿPᵣ = 840, ⁿCᵣ =35 then n, r is
a) 7,7 b) 7,4 c) 1,2 d) 2,1
114) x+y+z= 20, then how many ways it can be solved that x, y, z are not negative ...
a) ²⁰C₂ b) ²¹C₂ c) ²²C₂ d) None
115) In how many ways7 ′+′ signs and 5 ′ -′ sign can be arranged arranged in a line line without keeping any two ′ - ′ sign side by side..
a) 24 b) 75 c) 56 d) 106
116) In any polygon, 3 diagonals are not concurrent. If the total number of intersection within the polygon is 70, the number of diagonals will be..
a) 20 b) 10 c) 15 d) 5
117) ⁿ⁺²C₈ is to ⁿ⁻²C₄= 57 is to 16, the value of n will be...
a) 21 b) 19 c) 17 d) 11
118) If ¹⁵Pᵣ₋₁ is to ¹⁶Pᵣ₋₂ = 3 is to 4, then the value of r is..
a) 14 b) 20
c) 21. d) a,c right , b wrong
119) If permutation is it carried out taking for objects, out of 10 different objects, together in such a way that there always must be a special objects, the number of permutation will be..
a) 2016 b) 200 c) 1034 d) 1104
120) ᵐᵣ₌₀∑ ⁿ⁺ʳCₙ is equal to
a) ⁿ⁺ᵐ⁺¹Cₙ₊₁ b) ⁿ⁺ᵐ⁺²Cₙ
c) ⁿ⁺ᵐ⁺²Cₙ d) none
121) Out of 10 red and 8 white balls, 5 red and 4 white balls can be drawn in number of ways
A) ⁸C₅. ¹⁰C₄. B) ¹⁰C₅.⁸C₄ C)¹⁸₉ D) N
122) A polygon has 44 diagonals, then the number of its sides are
A) 11. B) 7. C) 8. D) none
123) If 7 points out of 12 are in the same straight line, then the number of triangles formed is:
A) 19. B) 185. C) 201. D) none
124) All the letters of the word EAMCET are arranged in all possible ways. The number of such arrangements in which no two vowels are adjacent to each other is:
A) 360. B) 144. C) 72. D) 54
125) 7 men and 7 women are sit round a table so that there is a man on either side of a woman. The number of seating arrangements is:
A) (7!)². B) (6!)². C) 6! .7! D) 7!
126) Total Number of words formed by the 2 vowels and 3 consonants taken from 4 vowels and 5 consonants is
A) 24. B) 1680. C) 7200. D) 72
127) The number of off numbers between 1000 and 10000 that can be formed with the digits 1,2,3,4,5,6,7,8,9 is..
A) 24. B) 1680. C) 7200. D) 72
128) Number of all four digits numbers having different digits formed of the digits 1,2,3,4,5 and divisible by 4 is
A) 24. B) 1680. C) 7200. D) 72
129) The number of proper divisors of 2⁶.3⁵. 5³. 7⁴ . 11 is
A) 1680. B) 144. C) 806. D) n
130) There are 18 points in a plane out of which 5 are collinear. Number of straight lines that can be formed is
A) 1680. B) 144. C) 806. D) n
131) Number of triangles that can be formed in Question 11
A) 1680. B) 144. C) 806. D) n
132)
A) The number of permutations containing the word ENDEA is
A) 5! B) 2. 5! C) 7.5! D) 21.5!
B) The number of permutations in which the letter E occurs in the first and the last position is
A) 5! B) 2. 5! C) 7.5! D) 21.5!
C) The number of permutations in which none of the letters D, L,.N, occurs in the last five position is
A) 5! B) 2. 5! C) 7.5! D) 21.5!
D) The number of permutations in which the letters A, E, O occur only in odd position is:
A) 5! B) 2. 5! C) 7.5! D) 21.5!
133) ⁿCᵣ₋₁= 56, ⁿCᵣ= 28 and ⁿCᵣ₊₁ = 8 then r is equal to
A) 8. B) 6. C) 5. D) none
134) The value of ⁴⁰C₃₁+ ¹⁰ⱼ₌₀∑⁴⁰⁺ʲ₁₀₊ⱼ is equal to
A) ⁵¹C₂₀ B) 2 .⁵⁰C₂₀
C) 2. ⁴⁵C₁₅ D) none
135) in a a group of boys, the number of arrangements of 4 boys 12 times the number of arrangements of 2 boys. The number of boys in the group is..
A) 10. B) 8. C) 6. D) none
136) the value of ¹⁰ⱼ₌₁∑ j. ʲPⱼ is..
A) ¹¹P₁₁.B) ¹¹P₁₁- 1.C) ¹¹P₁₁+1 D) n
137) From a group of persons the number of ways of selecting 5 persons is equals to that of 8 persons. The number of person in the group group is ..
A) 12 B) 40 C) 18 D) 21
138) The number of distinct rational numbers x such that 0 <x<1 and x= p/q, where p, q belongs to {1,2,3,4,5,6}, is
A) 15 B) 13 C) 12 D) 11
139) the total number of 9 digit numbers of different digits is..
A) 10. 9! B) 8. 9!
C) 9. 9! D) none
140) The number of 6 digit numbers that can be made with the digits 0 ,1, 2 ,3 ,4 and 5 so that even digits occupy odd places, is
A) 24. B) 36. C) 48. D) none
141) The number of ways in which 6 men can be arranged in a row so that 3 particular men are consecutive is
A) 24. B)144. C) 36. D) none
142) 7 different lectures are to deliver lectures in seven periods of a class on a particular day. A, B and C are 3 of the lectures. The number of ways in which a routine for the day can be made such that A delivers his lecture before B, and B before C is.
A) 420. B) 120. C) 210. D) none
143) The total number of Five digit numbers of different digits in which the digit in the middle is the largest is...
A) ⁹ₙ₌₄∑ⁿP₄ B) 33.3! C) 30. 3! D) N
144) A 5 digit number divisible by 3 is to be formed using the digits 0,1, 2, 3 ,4 and 5 without repetition. The total number of ways in which this can be done is..
A) 216. B)600. C)240. D)3125
145) let A={x|x is a prime number and x< 30}. The number of different ratii numbers whose numerator and denominator belong to A is
A) 90. B) 180. C) 91. D) none
146) The total number of ways in which six '+' and four '-' signs can be arranged in a line such that no two '-' signs occur together is..
A) 7!/3! B) 6!. 7!/3! C) 35. D) N
147) The total number of words that can be made by writing the letters of the word PARAMETER so that no vowel is between two consonants is...
A)1440 B)1800. C) 2160. D) N
148) The number of numbers of four different digits that can be formed that can be formed from the digits of the number 12356 such that the number are divisible by 4 is..
A) 36. B) 48. C) 12. D) 24
149) Let S be the set of all functions from the set A to the set A. If n(A)= k then n(S) is..
A) k! B) kᵏ C) 2ᵏ - 1. D) 2ᵏ
150) Let A be the set of 4 digit numbers a₁a₂a₃a₄ where a₁ >a₂ >a₃> a₄ then n(A) is equal to..
A) 126 B) 84 C) 210 D) none
151) The number of numbers divisible by 3 that can be formed by four different even digits is
A) 18. B) 36. C) 0. D) none
152) The number of 5 digit even numbers that can be made with the digits 0, 1, 2 and 3 is..
A) 384. B)192. C) 768. D) none
153) The number of 4 digit numbers that can be made with the digits 1,2,3,4,5 in which atleast two digits are identical, is..
A) 4⁵- 5! B) 505. C) 600. D) none
154) The number of words that can be made by rearranging the letters of the word APURBA so that vowels and consonants alternate is..
A) 18. B) 35. C) 36. D) none
155) The number of words that can be made by writing down the letters of the word CALCULATE such that each word starts and ends with the consonants, is
A) 5(7!)/2 B) 3(7!)/2
C) 2(7!) D) none
156) The number of a numbers of 9 nonzero digits such that all the digits in the first four places are less than the digit in the middle and all the digits in the last four places are greater than that in the middle is..
A) 2(4!). B) (4!)². C) 8! D) none
157) In the decimal system of numeration the number of 6 digit numbers in which the digit in any place is greater than the digit to the left of it is..
A) 210 B) 84 C) 126 D) none
158) The number of 5 digit numbers in which no two consecutive digits are identical is..
A) 9². 8³. B)9. 8⁴. C) 9⁵. D) none
159) In the decimal system of numeration the number of 6 digit numbers in which the sum of the digits is divisible by 5 is..
A) 180000.B) 54000 C) 5.10⁵ D) N
160) The sum of all the numbers of four different digits that can be made by using the digits 0, 1, 2 and 3 is..
A) 26664 B)39996 D)38664 D) N
161) A teacher takes 3 children from her class to the zoo at a time as often as she can, but she does not take the same three children to the zoo more than once. She finds that she goes to the zoo 84 times more than a particular child goes to the zoo. The number of children in her class each class is...
A) 12. B) 10. C) 60. D) none
162) ABCD is a convex quadrilateral 3, 4 ,5 ,6 points are marked on the sides AB, BC, CD DA respectively. The number of triangles with vertices on different sides is..
A)270. B) 220. C) 282. D) none
163) there are 10 points in a plane of which no three points are collinear and four points are concyclic. The number of different circles that can be drawn through at least 3 points of these points is..
A) 116 b) 120 C) 117 D) none
164) in a polygon the number of diagonals is 54. The number of sides of the polygon is..
A) 10. B) 12. C) 9. D) none
165) In a polygon no 3 diagonals are concurrent. If the total number of points of intersection diagonals interior to the polygon with 70 then the number of diagonals of the polygon is...
A) 20 B) 28 C) 8 D) none
166) n lines are drawn in a plane such that no two of them are parallel and no three of them are concurrent. The number of different points at which these lines will cut is...
A) ⁿ⁻¹ₖ₌₁∑ k B)n(n-1) C) n² D) none
167) The number of triangles that can be formed with 10 points as vertices, n of them being collinear is, 110. then n is..
A) 3 B) 4 C) 5 D) 6
168) there are three coplanar parallel lines. if any p points are taken on each of the lines, the maximum number of triangles with vertices at these points is..
A) 3p²(p-1)+1. B) 3p²(p-1)
C) p²(4p-3). D) none
169) Two teams are to play a series of 5 matches between them. A match ends in Win or loss or draw from a team. A number of people forecast the result of each match and no two people make the same forecast for the series of matc. The smallest group of people in which one person forcasts correctly for all the matches will contain n people, where n is..
A) 81. B) 243. C) 486. D) none
170) A bag contains 3 black, 4 white and 2 red balls, all the balls being different, The number of selections of atmost of 6 balls containing balls of all the colour
A) 42(4!). B)2⁶.4! C)(2⁶-1)4! D) n
171) In a room there are 12 bulbs of the same wattage, each having a separate switch. The number of ways to light the room with the different amount of illumination
A) 12²-1. B)2¹². C) 2¹² -1. D) n
172) In an examination of 9 papers a candidate has to pass in more papers than the number of papers in which he fails in order to be successful. The number of ways in which he can be unsuccessful is..
A) 255 B)256 C) 193 D) 319
173) The number of 5 digit numbers that can be made using the digit 1 and 2 and in which at least one digit is different is..
A) 30. B) 31. C) 32. D) none
174) In a club election the number contestant is one more than the number of maximum candidates for which a voter can vote. if the total number of ways in which a voter can go vote be 62 then the number of candidate is.
A) 7 B) 5 C) 6 D) none
175) The total number of selections of at most n things from different things (2n+1) different things is 63. Then the value of n is
A) 3 B) 2 C) 4 D) none
176) let 1≤ m < n ≤ p. The number of subsets of the set A={1,2,3..,p} having m, n as the least and the greatest elements respectively, is
A)2ⁿ⁻ᵐ⁻¹- 1 B) 2ⁿ⁻ᵐ⁻¹
C) 2ⁿ⁻ᵐ D) none
177) The number of ways in which n different prizes can be distributed among m(<n) persons if each is entitled to receive at most n-1 prizes, is..
A) nᵐ -n. B) mⁿ C) mn. D) none
178) The number of possible outcomes in a throw of n ordinary dice in which at least one of the dice shows an odd number is ..
A) 6ⁿ-1 B) 3ⁿ -1 C) 6ⁿ - 3ⁿ D) non
179) The number of different 6 digit numbers that can be formed using the three digits 0, 1 and 2 is
A) 3⁶. B) 2. 3⁵. C) 3⁵. D) none
180) The number of different matrices that can be formed with elements 0 ,1, 2 or 3, each Matrix having four elements is
A) 3. 2⁴. B) 2. 4⁴. C) 3.4⁴. D) none
181) let A be a set of n(≥3) distinct elements. The number of triplets (x,y,z) of the elements of A in which at least 2 coordinates are equals to to equals are equal is.
A) ⁿP₃ B) n³- ⁿP₃
C) 3n² - 2n D) 3n²(n-1)
182) The number of different pairs of words that can be made with the letters of the word (∆∆∆∆, ∆∆∆) that can be made with the letters of the word STATISTICS is..
A) 828 B) 1260 C) 396 D) none
183) Total number of 6 digit numbers in which all the odd digits and only odd digits appear, is
A)5.6!/2 B) 6! C)6!/2 D) none
184) The number of divisors of the form 4n+2(n≥2) of the integer 240 is.
A) 4 B) 8 C) 10 D) 3
185) In the next world cup cricket there will be twelve teams, divided equally in two groups. Teams of each group will play a match against each other. From each group 3 top teams will qualify for the next round. In this round each team will play against others once. 4 top teams of this round will qualify for the semi final round, where each team will play against the other once. Two top teams of this round will go to the final round, where they will play the best of three matches. The minimum number of matches in the next world cup will be.
A) 54. B) 53. C) 38. D) none
186) The number of different ways in which 8 persons can stand in a row so that between two particular persons A and B there are always two persons, is
A)60(5!). B) 15(4!)(5!)
C) 4! . 5! D) none
187) Four couples( husband and wife) decide to form a committee of four members. The number of different committee that can be formed in which no couple finds a place is..
A) 10. B) 12. C) 14. D) 16
188) From 4 gentlemen and 6 ladies A committee of 5 is to be selected. The number of ways in which the committee can be formed so that gentlemen are in majority is..
A) 66. B) 156. C) 60. D) none
189) There are 20 questions in a question paper. If no two students Solve the same combination of Questions but solve equal number of questions then the maximum number of students who appeared in the examination is..
A) ²⁰C₉. B) ²⁰C₁₁ D)²⁰C₁₀ D) none
190) Nine hundred distinct n-digit positive numbers are to be formed using only the digits 2,5 and 7. The smallest value of n for which this is possible is..
A) 6. B) 7. C) 8. D) 9
191) The total number of integral solutions for (x ,y ,z ) such that x y z= 24 is..
A) 36 B) 90 C) 120 D) none
192) The number of ways in which the letters of the word ARTICLE can be rearranged so that even places are always occupied by consonants is..
A) 576. B)⁴C₃ . 4! C) 2. 4! D) n
193) A cabinet ministers consist of 11 ministers, one Minister being the chief minister. A meeting is to be held in a room having a round table and 11 chairs round it, one of them being meant for the chairman. The number of ways in which the ministers can take their chairs, the chief minister of occupying the Chairman's place, is..
A) 10!/2. B) 9! C) 10! D) none
194) The number of ways in which a couple can sit around a table with 6 with 6 guests if the couple take consecutive seats is..
A)1440. B) 720. C)5040. D) none
195) The number of ways in which 20 different pearls of two colours can be set alternatively on a necklace, there being 10 pearls of each colour, is..
A) 9! 10! B) 5(9!)² C) (9!)² D) n
196) If r>p>q, the number of different selections of p+q things taking r are at a time, where p things are identical and q other things are identical is..
A) p+q-r B) p+q-r+1
C) r-p-q+1. D) none
197) There are 4 mangoes, 3 apples, 2 oranges and 1 each of three other varieties of fruits. The number of ways selecting at least one fruit of each kind is..
A) 10! B) 9! C) 4! D) none
198) The number of proper divisors of 2ᵖ 6ˢ15ʳ is
A) (p+s+r)(s+t+1)(r+1)
B) (p+s+r)(s+r+1)(r+1)-2
C) (p+s)(s+r)r - 2 D) none
199) The number of Proper divisors of 1800 which are also divisible by 10, is..
A) 18. B) 34. C) 27. D) none
200) The number of odd proper divisor of 3ᵖ 6ˢ 21ʳ is
A)(p+1)(s+1)(r+1)-2
B) (p+s+r+1)(r+1)-1
C) (p+1)(s+1)(r+1)--1. D) none
201) The number of even proper divisor of 1008 is
A) 23 B) 24 C) 22 D) none
202) in a test there where in questions. in the test 2ⁿ⁻¹ students gave wrong answers to i questions where i=1,2,3,...., nn. If the total number of wrong answers given is 2047 then n is
A) 12. B) 11. C) 10. D) none
203) The number of ways to give 16 different things to three persons A, B, C. so that B gets one more than A and C gets more than B.
A) 16!/(4!. 5!. 7!). B) 4!. 5!. 7!
C) 16!/(3!. 5!. 8!) D) none
204) The number of ways to distribute 32 different things equally among 4 persons is..
A) 32!/(8!)³. B) 32!/(8!)⁴
C) 32!/4. D) none
205) if 3n different things can be equally distributed among 3 persons in k ways then the number of ways to divide the 3n things in 3 equal groups is..
A) k. 3! B) k/3! C) (3!)². D) none
206) In a packet there are m different books, n different pens and p different pencils. The number of selections of at least one article of each type from the packet is..
A) 2ᵐ⁺ⁿ⁺ᵖ -1.
B)(m+1)(n+1))(p+1) -1
C) 2ᵐ⁺ⁿ⁺ᵖ D) none
207) The number of 6 digit numbers that can be made with the digit 1, 2 ,3 and 4 and having exactly two pairs of digit is..
A) 480 B) 540 C) 1080 D) n
208) The number of words of four letters containing equal number of Vowels and consonants, repetation being allowed, is..
A) 105². B)210x243
C) 105x 243. D) none
209) The number of ways in which 6 different balls can be put into two boxes of boxes of different sizes so that no box remains empty is.
A)62. B) 64. C) 36. D) none
210) A shopkeeper sells three varieties of perfumes and he has a large number of bottles of the same size of each variety in his stock. There are five places in a row in his showcase. The number of different ways of displaying the three verities of perfumes in the showcase is..
A) 6 B) 50 C) 150 D) none
211) The number of arrangements of the letters of the word BHARAT taking 3 at a time is..
A) 72 B) 120 C) 14 D) none
212) The number of ways to fill each of the four cells of the table with a distinct natural number such that the sum of the numbers is 10 and the sums of the numbers placed diagonally are equal, is
A) 2! 2! B) 4! C) 2.(4!) D) none
213) The number of positive integral solutions of x+y+z= n, n belongs to N, n≥ 3, is..
A) ⁿ⁻¹C₂ B)ⁿ⁻¹P₂ C) n(n-1) D) none
214) The number of non negative integral solutions of a+b+c+d= n, n belongs to N, is..
A) ⁿ⁺³P₂. B){((n-1)(n+2)(n-3)}/6
C) ⁿ⁻¹Cₙ₋₄ D) NONE
215) The number of points (xyz) in space, whose Coordinates is a negative integer such that x+ y +z+ 12 =0 is
A) 385. B) 55. C) 110. D) none
216) If a, b, c are three natural numbers in AP and a+ b+ c =21 then the possible number of values of the ordered triplet (a,b,c) is
A) 15. B) 14. C) 13. D) none
217) if a, b ,c ,d are odd natural numbers such that a+b+c+d=20 then the number of values of ordered quadruplet (a,b,c,d) is
A)163. B) 455. C) 310. D) none
218) If x, y, z are integers and x≥0, y≥1, z≥2, x+y+z =15 the number of values of the ordered triplet (x,y,z) is.
A) 91 B) 454 C) ¹⁷C₁₅ D) none
219) If a, b ,c are c are positive integers such that a+b+c≤8 then the number of possible values of the ordered triplet (a,b,c) is
A) 84 B) 56 C) 83 D) none
220) The number of different ways of distributing 10 marks among three questions, each question carrying at least one mark, is
A) 72. B) 71. C) 36. D) none
221) The number of ways to give away 20 Apples to three boys, each boy receiving at least four Apple is .
A) ¹⁰C₈. B) 90. C) ²²C₂₀. D) none
222) The position vector of a point P is r= xi+yj+tk, where x belongs to N, y belongs to N, z belongs to N and a= i+j+k. If r. a= 10, the number of possible position of P is
A) 36 B) 72. C) 66. D) none.
223) If P= n(n²-1²)(n² -2²)(n²-3²)....(n² - r²), n> r, n belongs to N, then P is divisible by
A) (2r+2)! B) (2r-1)!
C) (2r+1)! D) none
224) If ⁿ⁺⁵Pₙ₊₁ ={11(n-1)}/2 .ⁿ⁺³Pₙ then the value of n is..
A) 7 B) 8 C) 6 D) 5
225) If ⁿC₄ ,ⁿC₅ and ⁿC₆ are in AP then n is...
A) 8. B) 9. C) 14. D) 7
226) The product of r consecutive integers divisible by
A) r. B) ʳ⁻¹ₖ₌₁∑ k C) r ! D) none
227) There are 10 bags B₁, B₂, B₃, ... B₁₀, which contains 21, 22,...30 different articles respectively. The total number of ways to bring out 10 articles from a bag is..
A) ³¹C₂₀- ²¹C₁₀ B) ³¹C₂₁
C) ³¹C₂₀ D) None
228) If the number of arrangements of n - 1 things taken from n different things is k times the number of arrangements of n -1 things taken from n things in which two things are identical then the value of k is..
A) 1/2 B) 2 C) 4 D) now
229) Kanchan has 10 friends among whom two are married to each other. She wishes to invite 5 of them for a party. if the married couple refuses to attend separately then the number of different ways in which she can invite 5 friends is..
A) ⁸C₅ B) 2x ⁸C₃ C) ¹⁰C₅ - 2x⁸C₄ D) none
230) In a plane there are two families of lines y= x+r, y= - x+r, where r belongs to { 0,1,2,3,4}. The number of squares of diagonals of the length 2 formed by the lines is ..
A) 9. B) 16. C) 25. D) none
231) There are n seats round a table numbered 1, 2, 3, ....., n. The number of ways in which m(≤n) persons can take seats is..
A)ⁿPₘ. B) ⁿCₘ x +m-1)!
C) ⁿ⁻¹Pₘ₋₁ D)ⁿCₘ x m!
232) Let a= i+j+k and let r be variable vectors such that r. i, r.j and r.k are positive integers. If r. a ≤ 12 then the number of values of r is..
A) ¹²C₉ - 1. B) ¹²C₃ C) ¹²₉. D) non
233) The total number of ways in which a beggar can be given at least 1 Rupee from four 25 price coins, three 50 paise coins and 2 one-rupee coins, is..
A) 54. B) 53. C) 51. D) none
234) For the equation x+y+z+w= 19, the number of positive integeral solutions is equal to
A) the number of ways in which 15 identical things can be distributed among 4 persons.
B) the number of ways in which 19 identical things can be distributed among four persons.
C) Coefficient of x¹⁹ in (x⁰+ x¹ +x² +.... +x¹⁹)⁴
D) coefficient of x¹⁹ in (x+ x² + x³ +......+x¹⁹)⁴
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