1) How many odd numbers of 6 digits can be formed with the digits 0, 1, 4, 5, 6, 7, none of the digit being repeated in any numbers ? 288
2) From 10 different 6 things are taken at a time so that a particular thing is always included. Find the number of such permutations . 90720
3) How many different arrangements of the letters of the word FAILURE can be made so that the vowels are always together ? 576
4) In how many ways can the letters of the word STATION be arranged so that the vowels are always together ? 360
5) How many words can be formed by the letters of the word PEOPLE taken all together so that the two Ps are not together ? 120
6) In how many ways 6 books be arranged on a shelf of an almirah so that 2 prticular books will not be together? 480
7) Find the values of n and r when ⁿPᵣ = 336 and ⁿCᵣ = 56. 8,3
8) A box contains 10 electric lamps of which 4 are defective . Find the number samples of 6 lanpa taken at random from the box which will contain two detective lamps. 90
9) A committee of five is to be formed from 6 boys and 4 girls. How many different committees can be formed so that each committee contains at least two girls ? 186
10) An examinee has to answer 6 questions out of 12 questions . The questions are divided into two groups , each group containing 6 questions. The examinwe is not permitted to answer more than 4 questions from any group. In how many ways can he answer in all 6 questions ? 850
11) The Indian cricket eleven is to be selected out of fifteen players, five of them are bowlers. In how many ways the team can be selected so that team contains at least three bowlers? 1260
12) In a plane 5 points out of 12 points are collinear, no three of the remaining points are callinear. Find the number of straight lines formed by joining these points. 57.
13) Find the values of n and r, when ⁿPᵣ = ⁿPᵣ₊₁ and ⁿCᵣ = ⁿCᵣ₋₁. 3,2
14) which polygon has the same number of diagonals as sides ? Pentagon
15) How many words can be made out of the letters of the word EQUATION taken all at a time such that each word will have one constant in the beginning and one at the end. 4320
16) How many numbers of 7 digits can be formed with the digit 1,2, 3,4, 3, 2, 1 so that the odd digits will occupy odd places. 18
17) How many numbers greater than 1 lakh can be formed with the digits 0, 2, 5, 2, 4, 5 ? 150
18) In how many telephone numbers of 6 digits, two consecutive digits will be different? 9⁶
19) m men and n women take seats in a row. If m > n and no two women sit together then show that they can take seats in m!(m+1)!/(m - n +1)! ways.
20) Two person A and B sit with other 10 persons in a row. In how many arrangements will there be three persons in between A and B ? 16. 10!
21) Find the rank of the word MOTHER when its letters are arrange as in a dictionary. 309
22) 18 guests have to be seated, half on each side of a long table. 4 particular guests desire to sit on one particular side and three others on other side. Determine the number of ways in which the sitting arrangementa can be made. 2 x ¹¹C₅ x (9!)²
23) Show that the total number of permutations of n different things taking and more than r things at a time (repetition is allowed ) is n(nʳ -1)/(n -1).
24) If ⁿCᵣ₋₁/a = ⁿCᵣ/b = ⁿCᵣ₊₁/c, then show that,
n= (ab + 2ac + bc)/(b²- ac) and r= a(c + b)/(b²- ac)
25) If ⁿ⁺¹Cₘ₊₁ : ⁿ⁺¹Cₘ : ⁿ⁺¹Cₘ₋₁ = 5:5:3, find the values of m and n. 3,6
26) ⁴ⁿC₂ₙ/²ⁿCₙ = {1.3.5....(4n -1)}/{1.3.5....(2n -1)²}.
27) A committee of five persons is to be formed from 6 ladies and 4 gentlemen. How many committees containing at least two ladies can be formed where Mr A and Mrs B will
a) always remain
b) never remain. 55,56
28) Different arrangements are made by taking 3 vowels and 5 consonants out of 5 vowels and 10 consonants respectively so that the vowels always come together. Find the number of such arrangements. 10886400
29) In how many ways 4 or more men from 10 men can be selected ? 848
30) A box contains two white balls , 3 black balls and 4 red balls. In how many ways can three balls be drawn from the box if atleast one black ball is to be included in the draw ? 64
31) Find the total number of combinations taking at least one green ball and one blue ball from 5 different green balls, 4 different blue balls and 3 different red balls. 3720
32) A student is allowed to select atmost n books from a collection of (2n +1) books. If the total number of selections of at least one book is 63, then find the value of n. 3
33) A student has to select even number of books from a collection of 2n books. if he can select books in 2047 different ways, find the value of n. 6
34) How many different selections can be done taking at least one letter from each of the words TABLES , CHAIR, BENCH (2⁵-1)(2⁵-1)2⁵-1)= 29791
35) 2n out of 3n articles are alike and the rest are different. In how many ways 2n articles out of 3n articles can be selected. 2ⁿ
36) A box contains 2 guavas , 3 orange ps, 4 apples of different shapes.
a) in how many ways one or more fruits can be selected ?
b) how many selections of fruits can be made taking at least one of each kind? 511, 315
37) A question paper contains 10 questions . Four answers for each question are given of which one is correct. If one examinee gives answers for all of the 10 questions and select one answers for each question then in how many ways he can give correct answers for 5 questions? 61236
38) If any 7 dates are named at random then in how many cases of them, there will be three Sundays . 45360
39) There are two women participating in a chess competition. Every participant played 2 games with other participants. The number of games the men played between themselves exceed by 66 the number the meen played with the women . How many participated in the tournament and how many games were played? 13, 156
40) 5 balls of different colours that we placed in three boxes of different sizes All the five balls can be placed in each box. In how many different ways the balls can be kept in the boxes so that no box will be empty ? 150
41) In how many ways 12 different fruits can be divided among three boys so that each one can get at least one fruit ? 519156
42) There are four balls of four colours and 4 boxes of same colours with the balls. In how many different ways each box will contain one ball so that no ball will go to the box if its own colour. 9
43) a box contain 5 pair of shoes. In how many different ways can 4 shoes be selected so that there will be no complete pair of shoes? 80
44) A, B, C have 5,3 and 7 books of different types respectively. In how many different ways can they exchange the books so that number of books of everyone remains as before? 15!/(5!3!7!)
45) How many a) selections and b) arrangement can be made taking 4 letters from the word PROPORTION ? 53,758
46) How many numbers of 4 digit can be formed with the digits 1,1,2, 2, 3, 3, 4, 5 ? 354
47) Find the number of arrangements taking 5 letters at a time from the letters a,a,a,b,b,b,c,d. 320
48) How many different arrangements can be made by taking Four Pens from 3 same red colour pens, 2 same blue colours pens and 3 pens of other different colour ? 286
49) No 3 diagonals of a decagon are concurrent except at the vertices . Find the number of points of interaction of the diagonals. 595
50) AB and CD are two parallel straight lines. 20 points on AB and 20 points in CD are taken and the points on AB are connected with the points on CD . if no two straight lines are parallel and no three points are concurrent then how many Triangles can be formed so that one of the angular points of each triangle lies on AB and one on CD and another vertex does not lie on AB or CD? 31920000
51) How many Triangles can be obtained by joining the vertices of a polygon of 24 sides so that the sides of the triangle will not be the sides of the polygon ? 1520
52) The length and breath of a parallelograms are cut by p lines being parallel to both length and breadth . Show that in all (1/4)(p+1)²(p+2)² parallelograms are formed.
53) In a place there are m parallel roads along north-south and n parallel roads along east -west. In how many different short routes can a man go from the junction of north- east to the junction of Southwest? ⁿ⁺ᵐ⁻²Cₙ₋₁ (m+ n -2)!/{(n -1)!(m-1)!}
54) Find the number of different squares those can be formed on a chessboard. 204
55) There are n letters and n directed envelopes. In how many ways could all the letters be put into wrong envelopes ? n! (1/2! - 1/3! + 1/4! - .....+ (-1)ⁿ/n!)
56) In how many ways can 3A's, 2B's and 1C's be arranged in one line so that 2A's never occur together? 12
57) The number of 5 digit telephone numbers , none of their digits being repeated is
a) 50 b) ⁵⁰P₅ c) 5¹⁰ d) 10⁵
58) The number of 10 digit numbers formed with the digit 1 and 2, is
a) ¹⁰C₁ + ⁹C₂ b) 2¹⁰ c) ¹⁰C₂ d) 10!
59) A, B, C, D and E have been asked to deliver a lecture in a meeting. In how many ways can their lectures be arranged so that C delivers lecture just before A? 24
60) How many different signals can be given by using any number of the flags from six flags of different colours ? 1956
61) The number of ways in which five unlike rings a man can be wear on the four fingers of one hand
a) 120 b) 625 c) 1024 d) none
62) Show that the product of r successive natural number is divisible by r!.
63) If n> 7, prove that ⁿ⁻¹C₃ + ⁿ⁻¹C₄ > ⁿC₂.
64) The value of ⁴⁷C₄ + ⁵ᵢ₌₁∑⁵²⁻ⁱC₃ is
a) ⁴⁷C₄ b) ⁵²C₃ c) ⁵²C₄ d) none
65) The value of ¹⁵C₁+ ¹⁵C₃ + ¹⁵C₅+....+ ¹⁵C₁₅ =
a) 15!16! b) 15.2⁸ c) 2¹⁴ d) 2¹⁵
66) In a football championship, there were played 153 matches. Every two teams playex one match with each other. The number of teams, participating in the championship is
a) 17 b) 18 c) 9 d) none
67) Everybody in a room shakes handa with everybody else. The total number of hand shakes is 66. The total number of persons in the room is
a) 11 b) 12 c) 13 d) 14
68) The number of ways to form a team of 11 players out of 22 players where 2 particular players are included and 4 particular players are never included in the team is
a) ¹⁶C₁₁ b) ¹⁶C₅ c) ¹⁶C₉ d) ²⁰C₉
69) The total number of factors of 1998 (including 1 and 1998), is
a) 18 b) 16 c) 12 d) 10
70) In how many ways can 9 different things be divided into 3 groups of 2, 3 and 4 things respectively ? 1260
71) In how many ways can 12 different things be divided equally into 4 groups ? 15400
72) Of the four numbers 25, 150 , 170, 210 which one is the number of diagonals of a polygon of 20 sides ? 170
73) If a polygon has 54 diagonals, find the number of sides of the polygon. 12
74) In how many ways can the result (win or loss, or draw) of 3 successive football matches be decided? 27
75) There are 10 electric bulbs in a hall. Each of them can be lightened separately. The number of ways for lightning the ball is
a) 10² b) 1023 c) 2¹⁰ d) 10!
76) How many quadrilaterals can be formed with the seven sides of the length 1cm, 2 cms, 3 cms, 4 cms, 5 cms, 6 cms and 7 cms ? 32
77) How many different algebraic expression can be formed by combining the letters a, b, c, d, e, f with the signs '+' and '-', all the letters taken together? 64
78) Find the total number of ways in which six '+' and four '-' signs can be arranged in a line such that no two '-' signs occur together. 35
1) If ⁿCᵣ₋₁ = 56, ⁿCᵣ = 28 and ⁿCᵣ₊₁ = 8 then r is equal to
a) 8 b) 6 c) 5 d) none
2) The value of ⁴⁰C₃₁ + ¹⁰ⱼ₌₀ ∑ ⁴⁹⁺ʲC₁₀₊ⱼ is equal to
a) ⁵¹C₂₀ b) 2. ⁵⁰C₂₀ c) 2. ⁴⁵C₁₅ d) none
3) In a group of boys, the number of arrangements of 4 boys is 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
4) The value of ¹⁰∑ᵣ₌₁ r. ʳPᵣ is
a) ¹¹P₁₁ b) ¹¹P₁₁ -1 c) ¹¹P₁₁ +1 d) none
5) From a group of persons the number of ways of selection 5 persons is equals to that of 8 persons . The number of person in the group is
a) 13 b) 40 c) 18 d) 21
6) 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
7) The total number of 9 digit numbers of different digits is
a) 10. 9! b) 8.9! c) 9. 9! d) none
8) 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
9) The number of ways in which 6 men can be arranged in a row so that 3 perpendicular men are consecutive, is
a) ⁴P₄ b) ⁴P₄ x ³P₃ c) ³P₃ x ³P₃ d) none
10) Seven different lecturers are to deliver lectures in 7 periods of a class on a particular day. A, B and C are three of the lecturers . The number of ways in which a routine for the day in the can be made such that A delivers his lectures before B, and B before C, is
a) 420 b) 120 c) 210 d) none
11) The total number of 5 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) none
12) A five digit number divisible by 3 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
13) let A=x : x is a prime number and x< 30}. The number of different rational numbers whose numerator and denominator belongs to A is
a) 90 b) 180 c) 91 d) none
14) The total number of ways in which 6 '+' and 4 '-' 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) none
15) The total number of words that can be made by writing the letters of the word PARAMETER so that no vowels is between two consonants is
a) 1440 b) 1800 c) 2160 e) none
16) The number of numbers are four different digits that can be formed from the digit of the number 1 2 3 5 6 such that the numbers are divisible by 4, is
a) 36 b) 48 c) 12 d) 24
17) let S be the set of all fractions 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
18) Let A be the set of the four digit numbers a₁a₂a₃a₄ where a₁ >a₂>a₃>a₄ then n(A) is equal to
a) 126 b) 84 c) 210 d) none
19) The number of numbers divisible by 3 that can be formed by 4 different even digits is
a) 18 b) 36 c) 0 d) none
20) The number of 5 digit even numbers that can be made with the digit 0, 1 , 2 and 3
a) 384 b) 192 c) 768 d) none
21) The number of 4 digit number that can be made with the digit 1, 2, 3, 4 and 5 in which at least two digits are identical is
a) 4⁵ - 5! b) 505 c) 600 d) none
22) 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 a constant is
a) 5.7!/2 b) 3.7!/2 c) 2.7! d) none
23) The number of numbers of 9 different digits such that all the digits in the first four places are less than the digits in the middle and all the digits in the last four places are greater than that in the middle is 2.4! b) (4!)² c) 8! d) none
24) In the decimal system of numeration the number of 6 digit numbers in which the digits in any place is greater than the digit to the left of it is
a) 210 b) 84 c) 126 d) none
25) the number of 5 digit numbers in which no 2 consecutive digits are identical is
a) 9² . 8³ b) 9x 8⁴ c) 9⁵ d) none
26) 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) 540000 c) 5 x 10⁵ d) none
27) The sum of all the numbers of four different digits that can be made by the using 0, 1, 2,and 3: is
a) 26664 b) 39996 c) 38664 d) none
28) A teacher take three children from her class to the zoo at time as often as she can, but she does not take the same three childrens 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 is
a) 12 b) 10 c) 60 d) none
29) ABCD is a convex quadrilateral 3, 4, 5 and 6 points are marked on the sides AB , BC, CD and DA respectively. The number of triangles with vertices on different sides is
a) 270 b) 220 c) 282 d) none
30) 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 3 points of these points is
a) 116 b) 120 c) 117 d) none
31) 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
32) in a polygon no 3 diagonal concurrent. If the total number of points of intersection of the diagonals interior to the polygon be 70 then the number of diagonals of the polygon is
a) 20 b) 28 c) 8 d) none
33) n lines are drawn in a plane such that no two of them are parallel and no 3 of them are concurrent. The number of different points at which these lines will cuts is
a) ⁿ⁻¹∑ₖ₌₁ k b) n(n -1) c) n² d) none
34) 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, 4, 5, 6
35) 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
36) Two teams are to play a series of 5 matches between them. A match ends in a way or loss or draw for a team. A number of peoples forecast the result of each match and no two people make the same forecast for the series of the matches. The smallest group of people in which one person forecast correctly for all the matches will contain n people, where n is
a) 81 b) 243 c) 486 d) none
37) A bag contain three black, 4 white and two red balls, all the balls bings different. The number of selections of atmost six balls containing balls of all the colours is
a) 42 (4!) b) 2⁶ x 4! c) (2⁶-1)(4!) d) none
38) 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 different amounts of illumination is
a) 12²-1 b) 2¹² c) 2¹²-1 d) none
39) In an examination 9 papers of candidates 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 unsuccessful is
a) 255 b) 256 c) 193 d) 319
40) 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
41) In a club election the number contestants 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 vote be 62 then the number of candidates is
a) 7 b) 5 c) 6 d) none
42) The total number of selection of atmost n things from (2n +1) different things is 63. Then the value of n is
a) 3 b) 2 c) 4 d) none
43) 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
44) The number of ways in which n different prizes can be distributed among m(<n) persons if each is entitled to receive atmost n -1 prizes ,is
a) nᵐ b) mⁿ c) mn d) none
45) The number of possible outcomes in a throw of n ordinary dies in which at least one of the dice shows an odd number is
a) 6ⁿ-1 b) 3ⁿ -1 c) 6ⁿ - 3ⁿ d) none
46) The number of different 6 digit numbers that can be formed using the three digit 0, 1, 2 is
a) 3⁶ b) 2x 3⁵ c) 3⁵ d) none
47) The number of different matrices that can be formedd with elements 0, 1, 2norn3, each Matrix are 4 element, is
a) 3x 2⁴ b) 2x 4⁴ c) 3 x 4⁴ d) none
48) 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 two co-ordinates are equal is
a) ⁿP₃ b) n³ - ⁿP₃ c) 3n² - 2n d) 3²(n -1)
49) The number of different pairs of words (_ _ _, _ _ _) that can be made with the letters of the word STATICS is
a) 828 b) 1260 c) 396 d) none
50) Total number of 6 digit numbers in which all the odd digits and only odd digit appear, is
a) 5(6!)/2 b) 6! c) 6!/2 d) none
51) The number of divisors form 4n +2(n≥ 0) of the integer 240 is
a) 4 b) 8 c) 10 d) 3
52) in the next World Cup of cricket there will be 12 teams , divided equally in two groups. Teams of each group will play a match against each other. From each group 3 top team will qualify for the next round. In this round each team will play against other once. Four top teams of this round will qualify for the semi final round, where each team will play against the others once. Two top teams of this round will go to the final round, where they will play the best of 3 matches. The minimum number of matches in the next world cup will be
a) 54 b) 53 c) 38 d) none
53) 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! x 5! d) none
54) four couples (husband and wife) decide to form a committee of four members . The number of different committees that can be formed in which no couple finds a place is
a) 10 b) 12 c) 14 d) 16
55) from four gentlemen and 6 ladies a community of 5 to be selected. The number of ways in which the committee can be formed so that is gentlemen are in majority is
a) 66 b) 156 c) 60 d) none
56) There are 20 questions in a question paper. If no two students solve the same combination questions but solve equal number of questions then then the maximum number of student who appeared who appeared in the examination is
a)²⁰C₉ b) ²⁰C₁₁ c)!²⁰C₁₀ d) none
57) 9 hundred distinct n-digit positive numbers are to 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
58) The total number of integral solutions (x, y, z) such that XYZ= 24 is
a) 36 b) 90 c) 120 d) none
59) The number of ways in which the letters of the word ARTICLE can be rearranged so that the even places are always occupied by consonants is
a) 576 b) ⁴C₃. 4! c) 2(4!) d) none
60) a cabinet ministers consist of 11 ministers, one ministers 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 occupying the chairman's place is
a) 10!/2 b) 9! c) 10! d) none
61) The number of a ways in which a couple can sit around a table with 6 guests if the couple take consecutive seats is
a) 1440 b) 720 c) 5040 d) none
62) 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! x 10! b) 5(9!)² c) (9!)² d) none
63) If r> p > q, the number of different selections of p+ q things taking r at a time, where p things are identical and q things are identical is
a) p+ q+ r b) p+ q- r +1 c) - p- q+ r +1 d) none
64) There are four mangoes , three apples, two oranges and one each of 3 other varieties of fruits. The number of ways of selecting at least one fruit of each kind is
a) 10! b) 9! c) 4! d) none
65) The number of proper divisors of 2ᵖ. 6ᑫ. 15ʳ is
a) (p+ q+1)(q+r+1)(r+1)
b) (p+ q+1)(q+r+1)(r+1) -2
c) (p+ q)(q+r)r -2
d) none
66) The number of proper divisors of 1800 which are divisible by 10, is
a) 18 b) 34 c) 27 d) none
67) The number of odd proper devisors of 3ᵖ. 6ᑫ. 21ʳ is
a) (p+1)(m+1)(n+1)-2
b) (p+ m+ n+1)(n+1) -1
c) (p+ 1)(m+1)(n+ 1)
d) none
68) The number of even proper devisors of 1008 is
a) 23 b) 24 c) 22 d) none
69) in a test there were n questions. In the test 2ⁿ⁻ⁱ students of wrong answers to i questions where i= 1, 2, 3,.....n. if the total number of wrong answers given is 2047 then n is
a) 12 b) 11 c) 10 d) none
70) The number of ways to give 16 different things to 3 persons A, B and C so that B gets one more than A and C gets two more than B, is
a) 16!/(4!5!7!)
b) 4!5!7!
c) 16/(3! 5!8!) d) none
71) 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
72) if 3n different things can be equally distributed among 3 persons in k ways then the number of ways to divide 3n things in 3 equal groups is
a) k x 3 b) k/3! c) (3!)ᵏ d) none
73) in a packet there are m different books, n different pens and p different pencils. The number of selections of at least one articles 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
74) The number of 6 digits number 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) none
75) The number of words of 4 letters containing equal number of vowels and consonants , repeatation being allowed , is
a) 105² b) 210 x 243 c) 105 x 243 d) none
76) The number of ways in which 6 different balls can be put in two boxes of different sizes so that no box remains empty is
a) 62 b) 64 c)!36 d) none
77) 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 3 varieties of perfumes in the showcase is
a) 6 b) 50 c) 150 d) none
78) The numbers of arrangement of the letters of the word BHARAT taking 3 at a time is
a) 72 b)!120 c) 14 d) none
79) The number of ways to fill each of the four cells of the table with a distinct natural numbers such that the sum of the number is 10 and the sums of the numbers placed diagnolly are equal is
a) 2!2! b) 4! c) 2. 4! d) none
80) in the figure, 4 digit numbers are to be formed by filling the places with digits . The number of different ways in which the places can be filled by digits so that the sum of the numbers forme is also a 4 digit number and in no place the addition will carrying is
a) 555⁴ b) 220 c) 45⁴ d) none
81) 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
82) The number of non negative integral solution 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
83) The number of points (x,y, z)) in space, whose each co-ordinate is a negative integer such that x+ y+ z+12=0, is
a) 385 bb) 55 c) 110 d) none
84) If a,b,c are 3 natural numbers in AP and a+ b+ c= 21, then the possible number of values of the order ped triplet (a, b, c) is
a) 15 b) 14 c) 13 d) none
85) if a,b,c,,d are odd natural numbers that a + b + C + d = 20 then the number of values of the ordered quadruplet (a,b,c,d) is
a) 165 b) 455 c) 310 d) none
86) if x, y, z are integers and x≥0, y≥1, z≥2, x+ y+ z= 15 then the number of values of the ordered triplet (x,y,z) is
a) 91 b)) 455 c) ¹⁷C₁₅ d) none
87) If a,b,c are positive integer 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
88) The number of different ways of the distributing 10 marks among 3 questions, each question carrying atleast one mark is
a) 72 b) 71 c) 36 d) none
89) The number of ways to give away 20 apples in 3 boys, each boy receiving at least 4 apples, is
a) ¹⁰C₈ b) 90 c) ²²C₂₀ d) none
90) The position vector of a point P is r= xi + yj+ zk, where x ,y, 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
**
91) If P= n(n²-1)²(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
92) ⁿ⁺⁵Oₙ₊₁ = 11(n -1)/2. ⁿ⁺³Pₙ then the value of n is
a) 7 b) 8 c) 6 d) 9
93) If ⁿC₄, ⁿC₅ and ⁿC₆ are in AP then n is a
a) 8 b) 9 c) 14 d) 7
94) The product of r consecutive integers is divisible by
a) r b) ʳ⁻¹ₖ₌₁∑ k c) r! d) none
95) There are 10 bags B₁, B₂, B₃, .....B₁₀, which contain 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
96) 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) none
97) Kanchana has 10 friends among whom two are married to each other. she wishes to invite five of them for a party. if the married couple refuse to attend separately then the number of different ways in which she can invite 5 friends is
a) ⁸C₅ b) 2x ⁸C₃ c) ¹⁰C₅ - 2 x ⁵C₄ d) none
98) In a plane there are two families of lines y= x+ r, y= - z+ r, where r belongs {0, 1, 2, 3, 4}. The number of squares of the diagonals of the length 2 formed by the lines is
a) 9 b) 16 c) 25 d) none
99) there are n seats round a table numbered 1, 2, 3,....n. The number 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!
100) Let a= i+ j + k and let r= r be a variable vector such that r.j, 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) ¹²C₉ d) none
101) The total number of ways in which a beggar can be given at least one rupee from four 25 paise coins, three 50 paise coins and 2 one rupee coins, is
a) 54 b) 53 c) 51 d) none
102) for the equation x+ y+ z+ w= 19, the number of positive integral solutions is equals 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
1b 2a 3c 4b 5a 6d 7c 8a 9b 10d 11d 12a 13c 14c 15b 16a 17b 18c 19b 20a 21b 22c 23a 24b 25b 26c 27a 28c 29b 30d 31c 32b 33a 34a 35c 36c 37b 38a 39c 40b 41a 42c 43a 44b 45d 46c 47b 48c 49c 50b 51a 52a 53b 54a 55d 56a 57c 58b 59c 60a 61c 62a 63b 64b 65c 66b 67a 68b 69a 70b 71a. 72b 73b 74a 75c 76b 77a 78c 79a 80d 81d. 82a 83b 84b 85c 86a 87a 88b 89c 90a 91a 92bc 93ac 94cd 95abc 96a 97b 98bc 99a 100ad 101bc 102a 103ad
The number of ways in which we can post 5 letters in 10 letter boxes 5010 we can post the first letters in 10 days the second letters in 10 ways the number of 5 digit telephone number saving at least one of the digits 20000 132469786 a class is 30 student the following prices are to be awarded to the student of this classes first and second in mathematics first and second in physics first in chemistry and first in biology affect involved the number of a ways in which can be done is 468100 is divisible by 4 distinct prime numbers a later lock consist of three rings marked with 15 different literates of independent numbers of waves in which it is possible to make unsuccessful attends to open the lock 480 to end is a product of three distinct prime number change is the product of disting prime numbers the value of let p and a prime numbers is that 23 the number of primes in the least the 10 digit 1 2 3 49 is 1234 4 dials are hold the number of possible outcomes in which at least one die through 129625 671 S8 contains element the number of substation which contains at the most elements is a set of containing elements is against choose in the numbers using so that Airtel election mein voter mein board per any number of candidates not greater than the number of to which using that end candidate 5 minutes are to be chosen the number of waves and over the bottom about at least one candidates is given by 637 638 639 640 the sum of all the five digit numbers that can be found using the digits 12345 repetition of digit not allowed is 366 8 years are number 1282 women's at three men who is to occupied one chair is first woman choose the cheer from among the chairman to poor then the man select from the remaining chair the number of possible originates the number of signals that can be generated by the using different colour blacks take any number of them state at a time 1956 1997 199 if the latest of the word ratchet or arranged in all possible ways and these words are written out as an dictionary then the rank of the word Rachit is 365 for 81 720 a 5 digit number is divisible by 3:40 using the numerical 01234 and 5 without repetition the total number of a ways in which can be done 216 246525the number of occasion mix double games can be arranged from among 9 married couples of no husband and wife play in the same game is 756 152324 the sum of the divisors is if an objects are arranged in a row then the number of a ways of selecting 3 of the subject so that note to of them are next to each other is the number of ways of selecting 10 balls out of the unlimited number of white red blue and green balls is 27084 to 826 86 the number of non negative integrated solution of where is a positive integer is in a certain taste there are in question in this taste students give wrong answers to at least Al questions there want to if the total number of wrong answers given is 2014 then in is equals to 10 11 12 13 the number of time that digit 3 will be written when the listing the integers from 1 to 59 300 271 3002 if any positive integer in the value of number of positive integral solution 30 age that are 3 piece of identical card red blue and green balls and each file contains at least 10 boys the number upgrass of selecting 10 balls AP twice as a many red balls has been was at to be selected isidhar Seth containing elements of state issues in the set is a constructed by replacing the elements a subset is again choose in the number of ways using p and q so that contains exactly one elementase the value of expression there are five different books and mathematics to different books in chemistry and four different books and pages the number of ways the parenting these books also that books of the same subject together is there a 15 points in a plane of which exactly the total number of permutation different things taken not more than at the time when each of thing replaced any number of times 5 balls of different colours are to be placed in three boxes of different size is box can hold all five the number we can play the ball in the boxes so that no box remains 150 word is the number of integers greater than 7000 that can be found with the digit 35789 repeated is 10% among themselves in which it can be done it was to speak before the largest integer for a week 35 is divisible by age if an is a positive integer the value of the number of natural numbers which are smaller than and which can be written by means of the digit 1 into each the number of positive integral solution of 3527 06 is denote the set of a subset of containing exactly three elements two person are arranged in road the number of ways of selecting 4% so that no 2% sitting next to each other was selected there a 15 seats in the first row of a cinema hall the torch man has the instruction that the seat number of 6 must be occupied the number of waves in which four seats of the first row can be allotted so that note to of them are constitute is the number of positive integral solution of the inquality permutation of the letters of the word Rakshit are arranged in a dictionary three numbers at digit01 and string having exactly with the no zero consecutive is assuming that the value of 12233 nn if mn+1 care selected the return down each pair consisting at one chosen from them n letters and the other from the letters then that least identical the greatest value of the product of consecutive negative integer is divisible by that I cannot exist 2 positive integer such that are positive integer than each of them is equal cannot be square of the positive integer's given a positive integer the exist and consecutive positive integers prime for each positive integers the number of ways in which 5 heads of different colours can form a necklace is 12 the greatest value of is double the greatest value of the equation I get one solution in the set of natural numbers from any set a 52 integer it is always possible to change to integers that the sum are difference it is divisible by 100 denote the largest integer less than the cost to than value of the number of non negativity well solution of the equation such that the number of integral solution of where is an arrangement of number 1 to 100 then the product is an even integers there are 5 balls of different colours in 5 boxes of the same colours of those of the balls the number of waves when is the boss 1 each in a box could be place so that no ball goes to the box of its own colour is 44 the number of positive integers from 1 to 10 the inclusive which are neither purpose to another is the sun that are 20% among whom are two brothers the number of waves and avoid we can arrange them around a circle so that there is exactly one person between the two brothers is 28 if an even a question paper is split into two parts part A and party conting 5 question part PS4 is question is party has an alternative student has to attend at least one question from which part find the number of ways in the student and attend the person AG setup containing and elements a substate of a issues in the saves is reconstructed by replacing the elements next subset is chosen and again the set is the reconstructed by replacing the elements in this way or chosen find the number of ways of choosing what can be the maximum population of a country in which she can two person have an identical set of teeth this regard the same size of the day only the positioning of the teeth in consideration also asoom that there is no person without person is more than 32 bit how many different car licence can be constructed if the licence contains electricity English alphabet followed by 3 digits number of repetition are allowed and repetition are not allowed if x is a set containing and elements and why this it containing elements how many functions are there from how many of these functions are 211 prove that is a divisible by 2 so that no 21 seat together so that the number of which they can tea party is invent from people to sites of a long table and chairs side are man with two seat on 1 particular side and as on the other in how many ways can be seated as on that your family consist of grandfather and Samsung daughters they have to visit in the grandfather refuses to have a grand child on either side of him in how many ways can you family we met to sit find the number of ways are wearing 20 white balls and the three balls and no problem 6 of the place in the square of the diagram given the diagram such that each row contain at least one and how many ways and can A man is 7 relatives for women three men his wife also have 7 relative womens and four men and how many ways can they in 3 women and three men so that three of them mens relative and three is wipes and how many ways can a pack of it two cars be divided equally in a code players in order and how many ways can you divide this cards into force just one carsuppose city as an parallel roads running east west and unpairal roots running not South how many rectangles are found with their sides along this roads in the distance between every consecutive Pier a parallel road is the same how many shortest possible route are there to go from one corner to the city to diagonally opposite corner a box contains two white three black and forehead was in how many ways can 3 drawn from the box is to be included in the draw there are two women participating in a cheese tournament every participant played two games with every other participant the member of the games that main to win the themsel prove to exceed by 66 compared to the number of gains the main credit how many participants where they are how many games were played Mr accident by his first wife and Mrs be X + 1 children by her first husband they married and her children in their own the whole family has 10 children that to children of the same time is do not fight find the maximum number of that can take place an own children Bhavana support different toys and the has 7 different toys find the number of ways in exchange goes from Delhi to Jaipur stops at 7:00 intermediate station 5% enter the train during the journey by different towards of the same class distributed if you do anything parallel in direction han maine solution are there formation where each is a positive integers using so that the number of solution app with each and non negative integer to give that show that the sum of the number in the RO of the array illustrated in the phone given figure and has expressed the grand total of the numbers in the array as a single binomial coefficient by adding up the numbers in the array in a different ways to use that how many triangles oriented the same way as the can seas connected read like the own Sim the great consist of Andrews express the number of triangles which can be seen the other way up at the sum of the binomial coefficient also obtain the value of that sum by considering the colouring ke out of the industrial the item using one of the two colours for each of them shows that find the number of ways in a 16 apples can be distributed having 4% so that each of them gets at least one Apple find the number of whole numbers found in the screen on a calculator which can be recognise the numbers with the unique current is it when there is inverter the greatest number that can be found on the screen of the calculator is 999 find the number of natural numbers which are smallest then and artificial by 3 and which can be written by means of the digit 0 1 and 2 suppose there are 6 points in a plane no 3 of which are collinear is point is connected to other by a line segment and this line sigmand is coloured either rid or blue so that the respective of the ways way colour the connecting line segments they are always triangle with the vertices of this points and all who sides of the same colour from the set 1 to 200 prove that them on the selected the exist to distinct yourself that one of them is multiple of the other at party there are more boys and girls heach boy tences with exactly two girls prove that there is at least one girl who answers with at least three boysfind the number of ways are putting 5 distincts on a four fingers of the left and ignore the difference in the size of the rings and the fingers find the number of ways are arranging are distinct objects into an distinct boxes find the number of ways of placing identical objects into distinct boxes in any set of 10 to digit numbers shows that there always exist to non empty distinctive subscepes such that the sum of the numbers is it was to sum of the numbers give combination argument for the following identities if I understand things are arranged in a circle so that the number of ways of selecting this of this train so that note to obtain next to each other is so that they are carnot exist 2 positive integers for which are in HP in how many ways can I committee of 5 women and 6 men beaches in front and 8 men star it is to serve on the community is a member there are points no one straight line and appoint on another state line none of them how many triangles can be found with this points about two packs 52 playing cards is apple together find the number of ways in which can be dealt 26 cards with that he does not get to cards of the same suit and same denomination show that the number of different selection of 5 letters from 5 as 4b 3 C2 and 17 so that the number of ways selecting and different things out of things up which are one kind in a like second kind and like and latest so that the number of a ways on which three things one shot of another sort and short can be divided between two persons given things in each how many integers between 1 and 1 lakh have some of the digit equals to waiting in an examination the maximum marks from which of these peoples is and that for 4th paper is to improve that number of a way then who is candidates can get 3 and marks is find the number of in triple solution to the equation that are in straight lines in a plane no to which are parallel and no 3 of which has through the same point if the points of intersection are joint so that the number of additional lines that introduced page mcs in the first row of a theatre of vagina are be occupied final number of arranging in person so that no 2% seat bicycle each person as exactly one never and one and openly 26 located symmetrical about the middle of the row at least one is empty
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