PERMUTATION AND COMBINATION
Definition:
Permutation: Each of the arrangements in a definite order which can be made by taking some or all of a number things is called a PERMUTATION
Combination: Each of the groups or selections which can be made by taking some or all of a number of things without reference to the order of the things in each group is called COMBINATION.
Fundamental Principle of Counting:
If an event can occur in 'm' different ways, following which another event can occur in 'n' different ways, then the total number of different ways of simultaneous occurrence of both events in a definite order is m x n. This can be extended to any number of events.
RESULTS :
a) A useful notation : n! = n(n -1)(n -2).....3.2.1;
n! (n -1)!
0! = 1!= 1;
(2n)!= 2ⁿ. n! [1.3.5.7.....(2n -1)!]
Note that factorials of negative integers are not defined.
b) If ⁿPᵣ denotes the number of permutations of n different things, taking r at a time, then
ⁿPᵣ = n(n -1)(n -2)....(n - r -1)= n!/(n - r)! Note that, ⁿPₙ = n!
c) If ⁿCᵣ denotes the number of combinations of n indifferent things taken r at a time, then
ⁿCᵣ = n!/{r!(n - r)! = ⁿPᵣ/r! where r ≤ n ; n ∈ N and r ∈ W.
d) The number of ways in which (m + n) different things can be divided into groups containing m and n things respectively is: (m + n)!/m!n! If m= n, the groups are equal and in this case the number of subdivision is (2n)!/n!n!2!; for in any one way it is possible to interchange the two groups without obtaining a new distribution. However, if 2n things are to be divided equally between two persons then the number of ways = (2n)!/n!n!.
e) Number of ways in which (m + n + p) different things can be divided into 3 groups containing m,n and P things respectively is (m+ n + p)!/m!n!p!, m≠ n ≠ p.
If m= n =thep then the number of groups= (3n)!/n!n!n!3!.
However, if 3n things are to be divided equally among three peoples then the number of
ways= (3n)!/(n!)³.
f) The number of permutation of n things taken all at a time when p of them are similar and of one type, q of them are similar and of another type, r of them are similar and of a third type and the remaining n- (p+ q + r) are all different is : n!/p!q!r!.
g) The number of circular permutations of n different things taken all at a time is; (n -1)!. If clockwise and anti-clockwise circular permutations are considered to be same, then it is (n -1)!/2.
Note: Number of circular permutations of n things when p alike alike and the rest different taken all at a time distinguishing clockwise and anti-clockwise arrangement is (n -1)!/p!.
h) Given n different objects, the number of ways of selecting at least one of them is, ⁿC₁+ ⁿC₂ + ⁿC₃+....+ ⁿCₙ = 2ⁿ -1. This can also be stated as the total number of combinations of n distinct things .
i) Total number of ways in which it is possible to make a selection by taking some or all out of p+ q+ r+.... things, where p are alike of one kind, q like of a second kind, r alike of third kind and so on his given by : (p+1)(q+1)(r+1)......-1.
j) Number of ways in which it is possible to burn make a selection of m+ n + p= N things, where p are alike of one kind, m alike of second kind and n alike of third kind taken r at a time is given by coefficient of xʳ in the expansion of
(1+ x + x²+....+ xᵖ)(1+ x + x²+...+xᵐ)(1+ x + x²+...+xⁿ).
Note:
Remember that coefficient of xʳ in (1- x)⁻ⁿ = ⁿ⁺ʳ⁻¹Cᵣ (n ∈ N). For example the number of ways in which a selection of four letters can be made from the letters of the word PROPORTION is given by coefficient of x⁴ in (1+ x + x²+ x³)(1+ x + x²)(1+ x + x²)(1+ x)(1+ x)(1+ x).
k) Number of ways in which n distinct things can be distributed to p persons if there is no restriction to the number of things received by men= pⁿ.
l) Number of waya in which n identical things may be distributed among p persons if each person may receive none, one or more things is ⁿ⁺ᵖ⁻¹Cₙ.
m) i) ⁿCᵣ = ⁿCₙ₋ᵣ ; ⁿC₀ = ⁿCₙ =1 ;
ii) ⁿCₓ = ⁿCᵧ => x = y or x+ y = n
iii) ⁿCᵣ + ⁿCᵣ₋₁ = ⁿ⁺¹Cᵣ.
n) Let N= pᵃ qᵇ rᶜ ..... where p, q, r....are distinct primes and a,b, c....are natural numbers then:
i) The total numbers of divisors of N including 1 and N is= (a+1)(b +1)(c +1)...
ii) The sum of these divisors of is
= (p⁰+ p¹+ p²+....+ pᵃ)(q⁰+ q¹+ q²+...+qᵇ)(r⁰+ r¹+ r²+...+ rᶜ)....
iii) Number of ways in which N can be resolved as a product of two
(1/2) (a+1)(b+1)(c+1).... If N is not a perfect square
Factor is=
(1/2) [(a+1)(b+1)(c+1)....+1] if N is a perfect square.
iv) Number of ways in which a composite number N can be resolved into two factors which are relatively prime(or co-prime) to each other is equals to 2ⁿ⁻¹ where n is the number of different prime factors in N.
o) Grid Problem and tree diagrams.
Dearrangement:
Number of ways in which n letters can be placed in n directed letters so that no letter goes into its own envelope is= n![1/2! - 1/3! + 1/4!+.....+(-1)ⁿ (1/n!)]
p) Some times it is difficult to decide whether a problem is on permutation or combination or both. Based on certain words/ phrases occurring in the problems, you can fairly decide its nature as per the following table:
Combination Permutation
Selection Arrangement
Distributed group is formed Standing in a line seated in a row
Committee Problem on digits
Geometrical Problems Problems on letters from a word
PERMUTATION AND COMBINATION (MIXED)
BOOSTER - A
1) The straight lines l₁, l₂, l₃ are parallel and lie in the same plane. A total of m points are taken on the line l₁ n points on l₂ and k points on l₃. How many maximum number of triangles are there whose vertices are at these points. ᵐ⁺ⁿ⁺ᵏC₃ - (ᵐC₃ + ⁿC₃ + ᵏC₃)
2)a) How many 5 digits numbers divisible by 3 can be formed using the digits 0, 1, 2, 3, 4,7 and 8 if each digit is to be used atmost once. 744
b) Find the number of 4 digit positive integers if the product of their digits is divisible by 3. 7704
3) There are two women participating in a chess tournament. Every participant played 2 games with the other participants. The number of games that the men played between themselves exceeds by 66 as compared to the number of games that the men played with the women. Find the number of participants and the total number of games played in the tournament. 13,156
4) All the seven digit numbers containing each of the digits 1, 2, 3, 4, 5, 6, 7 exactly once, and not divisible by 5 are arranged in the increasing order. Find the 2004ᵗʰ number in this list. 4316527
5) five boys and 4 girls sits in a straight line. Find the number of a ways in which they can be seated if two girls are together and the other two are also together but separate from the first 2. 43200
6) A crew of an 8 oar boat has to be chosen out of 11 men five of whom can row on stroke side only, four on the bow side only, and the remaining two on either side. How many different selections can be made ? 145
7) An examination paper consists of 12 questions divided into parts A and B.
Part A contains 7 questions and part B contains 5 questions . A candidate is required to attempt 8 questions selecting at least 3 from each part. In how many maximum ways can the candidate select the questions ? 420
8) In how many ways can a team of 6 horses be selected out of stud of 16, so that there shall be 3 out of ABCA'B'C', but never AA', BB' or CC' together. 960
9) During a draw of lottery, tickets bearing numbers 1, 2, 3, ...., 40, 6 tickets are drawn out & then arranged in the descending order of their numbers. In how many ways, it is possible to have 4ᵗʰ tickets bearing number 25. ²⁴C₂. ¹⁵C₃
10) Find the number of distinct natural numbers upto maximum of 4 digit and divisible by 5, which can be formed with the digit 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 each digit not occuring more than once in each number. 1106
11) The Indian cricket team with 11 players, the team manager, the physiotherapist and two umpires are to travel from the hotel where they are staying to the stadium where the test match is to be played. Four of them residing in the same town own cars, each a 4 seater which they will drive themselves. The bus which was to pick them up failed to arrive in time after leaving the opposite team at the stadium . In how many ways can they be seated in the cars ? In how many ways can they travel by these cars so as to reach in time, if the seating arrangement in each car is immaterial and all the cars reach the stadium by the same route. 12!; (11!. 4!)/((3!)⁴.2!)
12) There are n straight lines in a plane, no 2 of which parallel, and no 3 pass through the same point. Their point of intersection are joined. Show that the number of maximum fresh lines thus introduced is n(n-1)(n-2)(n-3)/6.
13) In how many ways can you devide a pack of 52 cards equally among 4 players. In how many ways the cards can be divided in 4 sets, 3 of them having 17 cards each and 4ᵗʰ with one card. 52!/(13!)⁴; 52!/3(17!)³
14) A Firm of Chartered Accountants in Mumbai has to send 10 clerks to 5 different companies, two clerks in each. Two of the companies are in Mumbai and the others are outside. Two of the clerks prefer to work in Mumbai while three others prefer to work outside. In how many ways can the assignment be made if the preferences are to be satisfied. 5400
15) A train going from Cambridge to London stops at 9 intermediate stations. 6 persons enter the train during the journey with 6 different tickets of the same class. How many different sets of ticket may they have had ? ⁴⁵C₆
16) How many arrangements each consisting of 2 vowels and two consonants can be made out of the letters of the word DEVASTATION. 1638
17) Find the number of ways in which the letters of the word could 'KUTKUT' can be arranged so that no two alike letters are together . 30
18) Find the number of words each consisting of three consonants and three vowels that can formed from the letters of the word CIRCUMFERENCE, In how many of these c's will be together. 22100; 52
19) There are five white, four yellow, three green, two blue one 1 red ball. The balls are all identical except for colour. These are to be arranged in a line in 5 places. Find the number of distinct arrangements. 2111
20) How many four digit numbers are there which contains not more than two different digits? 576
21) In how many 8 persons can be seated on a round table.
a) if two of them( say A and B) must not sit in adjacent seats. 5.6!
b) if four of the person are men and four ladies and if no two men are to be in adjacent seats. 3!.4!
c) If 8 persons constitute 4 married couples and if no husband and wife, as well as no two men, are to be in adjacent seats? 12
22) If flight of stairs 10 steps. A person can go up the steps one at a time, two at a time, or any combination of 1's and 2's. Find the total number of ways in which the person can go up the stairs. 89
23) If 3 committees has 1 vacancy which is to be filled from a group of 6 people. Find the number of a way the three vacancies can be filled if:
a) Each person can serve on atmost 1 committee. 120
b) there is no restriction on the number of committees on which a person can serve. 216
c) Each person can serve on atmost 2 committees. 210
24) How many 10 digit whole numbers satisfy the following property they have 2 and 5 as the digits, and there are no consecutive 2's in the number (i.e., any 2's are separated by at least one 5). 143
25) In how many other ways can the letters of the word MULTIPLE be arranged ;
a) Without changing the order of the vowels. 3359
b) keeping the position of each vowel fixed. 59
c) without changing the relative order.position of vowels and consonants. 359
26) 12 persons are to be seated at a square table, three on each side. 2 persons wish to sit on the North side and two wish to sit on the east side. One other person insists on occupying the middle seat (which may be on any side). Find the number of a ways they can be seated. 2!.3!.8!
27) How many integers between 1000 and 9999 have exactly one pair of equal digit such as 4049 or 9902 but not 4449 or 4040 ? 3888
28) Determine the number of paths from the origin to the point (9, 9) in the Cartesian plane which never pass through (5,5) in paths consisting only of steps going 1 unit North and 1 unit East. 30980
29) a) prove that: ⁿPᵣ = ⁿ⁻¹Pᵣ + r. ⁿ⁻¹Pᵣ₋₁.
b) If ²⁰Cᵣ₊₂ = ²⁰C₂ᵣ₋₃ find ¹²Cᵣ. 792
c) Find the ratio ²⁰Cₚ to ²⁵Cᵣ when each of them has the greatest value possible. 143/4025
d) Prove that ⁿ⁻¹C₃ + ⁿ⁻¹C₄ > ⁿC₃ if n > 7.
e) Find x if ¹⁵C₃ₓ = ¹⁵Cₓ₊₃. 3
30) There are 20 books on Algebra and calculus in our library. Prove that the greatest number of selections each of which consists of 5 books on each topic is possible only when there are 10 books on each topic in the library.
BOOSTER - B
1) Find the number of a ways in which 3 distinct numbers can be selected from the set {3¹, 3², 3³,.....3¹⁰⁰,3¹⁰¹} so that they form a GP. 2500
2) There are counters available in 7 different colours. Counters are all alike except for the colour and they are at least 10 of each colour. Find the number of ways in which an arrangement of 10 counters can be made. How many of these will have counters of each colour . 7¹0; (49/6) 10!
3) For each positive integer k, let Sₖ denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is k. For example, S₃ is the sequence 1,4,7,10.... find the number of values of k for which Sₖ contains the term 361. 24
4) Find the number of 7 lettered words each consisting of 3 vowels and 4 consonants which can be formed using the letters of the word DIFFERENTIATION. 532770
5) A shop sells 6 different flavours of icecream. In how many ways can a customer choose 4 ice cream cones if
a) they are all of different flavours. 15
b) they are not necessarily of different flavours . 126
c) they contains only three different flavours. 60
d) they contains only two or three different flavours? 105
6) There are n triangles of positive area that have one vertex A(0,0) and the other two vertices whose coordinates are drawn independently with replacement from the set {0,1,2,3,4} e.g., (1,2),(0,1)(2,2) etc. Find the value of n. 256
7) There are 2n guests at a dinner party. Supposing that the master and mistress of the house have fixed seats opposite one another, and that there are two specified guests who must not be replaced next to one another. Show that the number of ways in which the company can be placed is (2n -2)! (4n²- 6n +4).
8) a) How many divisors are there of the number x= 21600. Find also the sum of these divisors. 72; 78120
b) In how many ways the number 756 can be resolved as a product of two factors find the number of ways the number 7056 can be resolved as a product of 2 factors. 23
c) Find the number of ways in which the number 300300 can be split into 2 factors which are relative prime. 32
9) How many 15 letters arrangements of 5 A's, 5 B's and 5C's have no A's in the first five letters, no B's in the next 5 letters, and no C's in the last 5 letters. 2252
10) How many different ways can 15 Candy bars be distributed between Ram, Shyam, Ghanshyam and Balram, if Ram cannot have more than 5 candy bars and Shyam must have at least two . Assume all candy bars to be alike. 440
11) Find the number of ways in which the number 30 can be partitioned into 3 unequal parts, each part being a natural number. What this number would be if equal parts are also included. 61,75
12) In an election for the managing committee of a reputed club, the number of candidates consisting elections exceed the number of members to be elected by r (r > 0). if a vector can vote in 967 different ways to elect the managing committee by voting at least one of them and can vote in 55 different ways to elect (r -1) candidates by voting in the same manner. Find the number of candidates contesting the elections and the number of candidates losing the elecs. 10,3
13) Find the number of 3 digits numbers from 100 to 999 inclusive which have any one digit that is the average of the other two. 121
14) A man has 3 friends. In how many ways he can invite one friend everyday for dinner on 6 successive nights so that no friends is invited more than three times. 510
15) Find the number of distinct throws which can be thrown with the 'n' six faced normal dice which are indistinguishable among themselves. ⁿ⁺⁵C₅
16) There are 15 rowing clubs, two of the clubs have each 3 boats on the river; five others have each 2 and the remaining 8 have each one; find the number of ways in which a list can be formed of the order of the 24 boats, observing that the second boat of a club cannot be above the first and the third above the second. How many ways are there in which a boat of the club having single boat on the river is at the third place in the list formed above? 24!/((3!)²(2!)⁵); ⁸C₁. 23!/((3!)²(2!)⁵
17) Consider a 7 digit telephone number 336-7624 which has the properly that the first three digit profix, 336 equals the product of the last four digits. How many 7 digit phone number beginning with 336 have this property., e.g(336-7624). 84
18) An 8 oared boat is to be manned by a crew chosen from 14 men of which 4 can only steer but can not row and the rest can row but cannot steer. Of those who can row, 2 can row on the both side. In how many ways can the crew be arranged. 4.(4!)². ⁸C₄. ⁶C₂
19) How many 6 digits odd numbers greater than 600000 can be formed from the digit 5, 6, 7, 8, 9, 0 if
a) repetation are not allowed. 240
b) repetitioner are not allowed. 15552
20) Find the sum of all numbers greater than 10000 farmed by using the digits 0, 1, 2, 4, 5 no digits being repeated in any number. 3119976
21) The members of a chess club took part in a round robin competition in which each plays everyone else once. All members scored the same number points, except 4 juniors whose total score 17.5. How many members were there in the club ? Assume that for each win players scores 1 point, for draw 1/2 point and zero for losing. 27
22) There are three cars of different make available to transport 3 girls and 5 boys on a field trip. Each car can hold up to three children. Find
a) the number of a ways in which they can be accommodated . 1680
b) the numbers of ways in which they can be accommodated if two or three girls are assigned to one of the cars. In both the cases internal and arrangement of children inside the car is considered to be immaterial. 1140
23) Find the number of three elements sets of positive integers (a, b, c) such that a x c = 2310. 40
24) Find the number of integer between 1 and 10000 with at least one 8 and at least one 9 as digits. 974
25) In Indo-Pak one day international cricket match at Sharjah , India needs 14 runs to win just before the start of the final over. Find the number of ways in which India just manages to win the match (i.e., scores exactly 14 runs), assuming that all the runs are made off the bat and the batsman can not score more than 4 runs off any ball. 1506
26) A man goes in for an examination in which there are four papers with a maximum of m marks for each paper; Show that the number of ways of getting 2m marks on the whole is (1/3) (m+1)(2m²+ 4m +3).
BOOSTER - C
1) How many different 9 numbers can be found from the numbers 223355888 by rearrange its digit so that the odd digits occupy even positions ?
a) 16 b) 36 c) 60 d) 180
2) Let Tₙ denote the number of triangles which can be formed using the vertices of a regular polygon of n sides. If Tₙ₊₁ - Tₙ =21, then n equals:
a) 5 b) 7 c) 6 d) 4
3) The number of arrangements of the letters of the word BANANA in which the two N's do not appear adjacently is
a) 40 b) 60 c) 80 d) 100
4) Number of points with integral co-ordinates that lie inside a triangular whose coordinates are (0,0),(0,21) and (21,0)
a) 210 b) 190 c) 220 d) none
5) Using permutation or otherwise, Prove that (n²)!/(n!)ⁿ is an integer, where n is a positive integer.
6) A rectangle with side 2m -1 and 2n -1 is divided into squares of unit length by drawing parallel lines as shown in the diagram,
then the number of rectangles possible with odd side lengths is:
a) (m+ n+ 1)² b) 4ᵐ⁺ⁿ⁺¹ c) m²n² d) mn(m+1)(n+1)
7) If r, s, t are prime numbers and p, q are the positive integers such that their LCM of p, q, is r²t⁴s², then the numbers of order pair of (p, q) is
a) 252 b) 254 c) 225 d) 224
8) The letters of the word COCHIN are permuted and all the permutations are arranged in an alphabetic order as in an English dictionary. The number words that appear before the word COACHIN is
a) 360 b) 192 c) 96 d) 48
9) Consider all possible permutation of the letters of the word ENDEANOEL
Match the statements/expression
Column A
a) The number of permutations containing the word ENDEA is
b) The number of permutations in which the letter E occurs in the first and the last position is
c) The number of permutations is which none of the letters D, L, N occurs in the last five positions is
d) The number of permutations in which the letters A, E, O occurs only in odd positions is:
Column B
i) 5!
ii) 2 x 5!
iii) 7 x 5!
iv) 21 x 5!
10) The number of 7 digit integers, with sum of the digits equal to 10 and formed by using the digit 1, 2 and 3 only, is
a) 55 b) 66 c) 77 d) 88
11) Let S={1,2,3,4}. The total number of unordered pairs of disjoint sub-sets of S is equal to
a) 25 b) 34 c) 42 d) 41
12) The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets atleast one ball is
a) 75 b) 150 c) 210 d) 243
13) Let Tₙ be the number of all possible Triangle formed by joining vertices of an n-sided regular polygon. If Tₙ₊₁ - Tₙ= 10, then the value of n is
a) 8 b) 7 c) 6 d) 10
14)Let n₁ < n₂ < n₃ < n₄ < n₅ be positive integers such that n₁+ n₂+ n₃ + n₄ + n₅ = 20. Then the number of such distinct arrangements (n₁, n₂, n₃, n₄, n₅) is. 7
15) Let ≥ 2 be an integer. Take n distinct points on a circle and join each pair of points by a line segment . Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of n is . 5
16) Six cards and six envelopes are numbered 1, 2, 3, 4, 5, 6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same numbers and moreover the card numbered 1 is always placed in envelope numbered 2. Then the number of ways it can be done is:
a) 264 b) 265 c) 53 d) 67
1c 2b 3a 4b 6d 7c 8c 10c 11d 12b 13c 16c
BOOSTER - D
1) How many of the 900 three digit numbers have at least one even digit ?
a) 775 b) 875 c) 450 d) 750
2) The number of natural numbers from 1000 to 9999 (both inclusive) that do not have all four different digits is
a) 4048 b) 4464 c) 4518 d) 4536
OR
What can you say about the number of even numbers under the same constraints ?
3) The number of different 7 digit numbers that can be written using only 3 digits 1, 2 & 3 under the condition that the digit 2 occurs exactly twice in each number is
a) 672 b) 640 c) 512 d) none
4) Out of seven consonants and four vowels , the number of words of 6 letters, formed by taking four consonants and two vowels is (Assume that each ordered group of letter is a word).
a) 210 b) 462 c) 151200 d) 332640
5) All possible three digits even numbers which can be formed with the condition that if 5 is one of the digit, then 7 is the next digit is
a) 5 b) 325 c) 345 d) 365
6) Number of 5 digit numbers which are divisible by 5 and each number containing the digit 5, digits being all different is equals to k(4!), the value of k is
a) 84 b) 168 c) 188 d) 208
7) The number of six digit numbers that can be formed from the digits 1, 2, 3, 4, 5, 6, 7 so that the digits do not repeat and the terminal digits are even is
a) 144 b) 72 c) 288 d) 720
8) In a certain strange language, words are written with letters from the following six-letter alphabet: A,G, K, N, R, U. Each word consists six letters and none of the letters repeat. Each combination of these six letters is a word in this language. The word KANGUR remains in the dictionary at
a) 248th b) 247th c) 246th d) 253rd
9) Consider the five points comprising of the vertices of a square and the intersection point of its diagonals. How many triangles can be formed using these points?
a) 4 b) 6 c) 8 d) 10
10) A 5 digit number divisible by 3 is to formed using the numerals 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways this can be done is:
a) 3125 b) 600 c) 240 d) 216
11) Number of 9 digits numbers divisible by none using the digits from 0 to 9 If each digit is used atmost once is K. 8!, then K has the value equals to___.
12) Number of natural number less than 1000 and divisible by 5 can be formed with the ten digits, each digit not occurring more than once in each number is___.
13) Number of three digit numbers in which the digit at hundredth's place is greater than the other two digit is
a) 285 b) 281 c) 240 d) 204
14) Number of permutations of 1, 2, 3, 4, 5, 6, 7, 8 and 9 taken all at a time, such that the digit
1 appearing somewhere to the left of 2
3 appearing to the left of 4 and
5 somewhere to the left of 6, is
(e.g., 815723946 would be one such permutation)
a) 9.7! b) 8! c) 5!4! d) 8!4!
15) Number of odd integers between 1000 and 8000 which have none of their digits repeated, is
a) 1014 b) 810 c) 690 d) 1736
16) Find the number of ways in which letters of the word VALEDICTORY be arranged so that the vowels may never be separated
17) The number of ways in which 5 different books can be distributed among 10 people if each person can get atmost one book is?
a) 252 b) 10⁵ c) 5¹⁰ d) ¹⁰C₅. 5!
18) A new flag is to be designed with six vertical strips using some or all of the colours yellow, green, blue and red. Then, the number of ways this can be done such that no two adjacent strips have the same colour is
a) 12x 81 b) 16x192 c) 20× 125 d) 24× 216
19) 5 Indian & 5 American couples meet at a party & shake hands. If no wife shakes hands with her own husband & no Indian wife shakes hands with a male, then the number of hand shakes that takes place in the party is:
a) 95 b) 110 c) 135 d) 150
20) There are 720 Permutations of the digits 1,2,3,4,5,6. Suppose these Permutations are arranged from smallest to largest numerical values, beginning from 1 2 3 4 5 6 and ending with 6 5 4 3 2 1.
a) What number falls on the 124ᵗʰ position?
b) What is the position of the number 321546 ?
1a 2b or 2204 3a 4c 5d 6b 7d 8a 9c 10d 11) K=17 12) 154 13a 14a 15d 16) 967680 17d 18a 19c 20a) 213564 b) 267th
BOOSTER - E
1) How many numbers between 400 and 1000(both exclusive) can be made with the digit 2,3,4,5,6,0 if
a) repetition of digit not allowed
b) repetition a digits is allowed
2) The 9 horizontal and 9 vertical lines on 8 x 8 chessboard 'r' rectangles and 's' squares. The ratio s/r in its lowest term is
a) 1/6 b) 17/108 c) 4/27 d) none
3) A student has to answer 10 out of 13 questions in an examination. The number of ways in which he can answer if he must answer atleast 3 of the first five questions is:
a) 276 b) 267 c) 80 d) 1200
4) The number of three digit numbers having only two consecutive digits identical is:
a) 153 b) 162 c) 180 d) 161
5) A telegram has x arms & each arm is capable of (x -1) distinct positions , including the position of rest. The total number of signals that can be made is____.
6) The interior angles of a regular polygon measure 150° each . The number of diagonals of the polygon is
a) 35 b) 44 c) 54 d) 78
7) Number of different natural numbers which are smaller than two hundred million and using only the digit 1 or 2 is:
a) (3). (2⁸ -2) b) (3).(2⁸-1) c) 2(2⁹-1) d) none
8) The number of n digit numbers which consists of the digits 1 and 2 only if each digit is to be used at least once, is equals to 510 then n is equals to
a) 7 b) 8 c) 9 d) 10
9) Number of six digit numbers which have 3 digits even & 3 digits odd, if each digit is to be used at most once is ____
10) Find the number of 10 digit numbers using the digits 0,1, 2,.....9 without repetition. How many of these are divisible by 4.
11) There are counters available in x different colours. The counters are all alike except for the colour. The total number of arrangements consisting of y counters , assuming sufficient number of counters of each colour, if no arrangement consists of all counters of the same colour is:
a) xʸ - x b) xʸ - y c) yˣ - x d) yˣ - y
12) Number of four digit numbers with all digits different and containing the digit 7 is
a) 2016 b) 1828 c) 1848 d) 1884
13) An English school and a Vernacular school are both under one superintendent. Suppose that the superintendentship, the four teachership of English and Vernacular school each, are vacant, if there be altogether 11 candidates for the appointments, 3 of whom apply exclusively for the superintendentship and 2 exclusively for the appointment in the English school, the number of ways in which the different appointments can be disposed of is:
a) 4320 b) 268 c) 1080 d) 25920
14) A committee of 5 is to be chosen from a group of 9 people. Number of ways in which it can be formed if two particular persons either serve together or not at all and two other particular persons refuse to serve with each other, is
a) 41 b) 36 c) 47 d) 76
15) A question paper on mathematics consists of 12 questions divided into 3 parts A, B and C each containing four questions. In how many ways can an examinee answer 5 questions, selecting at least one from each part
a) 624 b) 208 c) 1248 d) 2304
16) If m denotes the number of fire digit numbers if each successive digits are in their descending order of magnitude and n is the corresponding figure, when the digits are in their assending order of magnitude then (m - n) has the value
a) ¹⁰C₄ b) ⁹C₅ c) ¹⁰C₃ d) ⁹C₃
17) There are m points on a straight line AB and n points on line AC none of them being the point A Triangle are formed with these points as vertices, when
i) A is excluded ii) A is included, The ratio of number of triangles in the two cases is :
a) (m+ n-2)/(m+ n)
b) (m+ n-2)/(m+ n -1)
c) (m+ n-2)/(m+ n+2)
d) m(n-2)/{(m+ 1)(n+1)}
18) In a certain algebraical exercise book there are 4 examples on arithmetic progression, 5 examples on permutation-combination and 6 examples on binomial theorem. Number of ways a teacher can select for his pupils at least one but not more than two examples for each of these sets is ____
19) Find the number of ways in which two squares can be selected from an 8 by 8 chess board of size 1 x 1 so that they are not in same row and in the same column.
20) The number of 5 digit numbers such that the sum of their digits is even is:
a)!50000 b) 45000 c) 60000 d) none
1a- 60 b-107 2b 3a 4b 5) (x -1)ˣ -1 6c 7a 8c 9) 64800 10) (20). 8! 11a 12c 13d 14a 15a 16b 17a 18) 3150 19) 1568 20b
BOOSTER - F
1) Number of ways in which 8 people can be arranged in a line if A and B must be added next each other and C must be somewhere behind D, is
a) 10080 b) 5040 c) 5050 d) 10100
2) Number of ways in which 7 green bottles and 8 blue bottles can be arranged in a bank row if exactly 1 pair of green bottles is side by side, is (Assume all bottle to be alike except for the colour)
a) 84 b) 360 c) 504 d) none
3) The kindergarten teacher has 25 kids in her class. She takes 5 of them at a time, to zoological garden as often as she can, without taking the same 5 kids more than once. Then the number of visits, the teacher makes to the garden exceeds that of a kid by:
a) ²⁵C₅ - ²⁴C₅ bb) ²⁴C₅ c) ²⁴C₄ d) none
4) seven different coins are to be divided among 3 persons if no to of the persons receive the same number of coins but h receives at least one coin and a none is leftover then the number of a ways in the way the division maybe maybe 420 630 710 let there be 9 fixed points on the circumference of a circle each of these points A joint to everyone of the remaining 8 points by a straight line in the points are so position on the circumference that atmos two straight line mid in any interior point of the circle the number of such interior intersection the number of ways in which 8 distinguishable apples can be distributed amount 3 boys such that every boy should get at least one apple and at the most four happens is where has the value equals to 1466 44 22 a woman has 11 close friends find the number of ways in which C can invite 5 of them to dinner if two particular up them are not on speaking terms and will not attend to pay a rack has 5 different periods of shoes the number of ways in which force shoes can be chosen from it so that there will be no complete fair is 1990 2280 there are 10 seats in double decker bus 6 in the lower dic and four on the upper day 10 passengers bored the bus of them three refused to go to the upper deck and 2 in system going up the number of west in which the passengers can be accommodate is Mall ship to be deuli number find the number of permutation of the word in which vowels appear in an alphabetical order there at 10 different books in a Cell find the number of waves and which three books can be selected so that exactly two of them are consecutive and old man while dialling is 7 digit telephone number that the first four digit consist of one he also recommendous that 5th digit is either a four or five while as a no memorizing of the 6 digit remembers that 7th digit is a 9 - 6 digit maximum number of distinct to try to make sure that the correct telephone number is 360 to 40 15
If has many more words as possible be formed out of the letters of the word then the number of words in which the relative order of Bhavesh and consonants remain and changed is number of awais in which 7 people can occupies 6 seats 3 seats on his side in a first class railway compartment to specified personnel to be always included and occupi at the same switch on the same side is the value equals to 248 number of ways in which 9 different toys be distributed among four children belonging to different age groups in such a way that distribution among the three welder children is even and the youngest one is to receive one toy more is there are five different features and three different apples number of ways they can be divided into two packs of 4 fruits which pack must container least one Apple 895 65 63
EXERCISE -G
1) How many 6 digits odd numbers greater than 600000 can be formed from the digits 5,6,7,8,9,0 if
a) repetitions are not allowed. 240
b) repetitions are allowed. 15552
2) 2 American men; 2 British men; 2 Chinese men and one each of Dutch, Egyptian, French and German persons are to be seated for a round table conference.
a) If the number of ways in which they can be seated if exactly two pairs of persons of same nationality are together is p(6!), then find p. 60
b) if the number of ways in which only American pair is adjacent is equal to q(6!), then find q. 64
c) If the number of ways in which no two people of the same personality are together given by R(6!), find r. 244
3) Find the number of positive integers that are divisors of atleast one of the number 10¹⁰; 15⁷; 18¹¹. 435
4)
The sum of all number support digit that can you made by using the digit 0123 the sum of all four digit number that can be found by using the digit 2468 when the petition of digit is not allowed the number of audit pairs of integers satisfying by provision among tempered persons are to speak at a function the number of ways in which it can be done it wants to speak before and to speak before in how many ways can a team of 11 players reforms out of 25 players to 6 out of them are always to be included in 5 always to be excluded the number of the ways in which the letters of the word person can be placed in the squares of the given figure so that no rows remain. The number of words of four letters that can be formed from the letters of the word the number of even divisor of the number in an election the number of candidate is one greater than the person to be elected if a motor can both in 254 ways the number of candidates a person predicts in the outcome of 20 cricket matches office only which match can result in either will lost or time for the home team total number of a ways in which he can make the operating so that exactly there are 10 points in a plane of which may 3 points are collinear and the Four points are from cyclic the number of different circles that can be drawn through at least 3 points of this points age there are three coplaner parallel lines if any points are technology at the lines the maximum number of bangles with vertices on this phone is the maximum number of points of intersection of five lines and the food circles is the number of interpreneur solutions the number of a ways in which 12 books number of Ho gayi then a weight 25 identical things with distributed among 5% secretariats odd numbers of things is the total number of divisors of 480 but area perform is equal to the number of three digits numbers of the phone such that MMS 3 friend the number of ways we can invite one friend everyday for dinner on 6 successive nights so that no friend is invited more than three times each a bag contains more Rs 1.225 price in 5 10 price points in how many ways can I amount not less than be taken out from the bag consider
There are 10 points in a plane of which no 3 points are coal in air but 4 points are from cycling the number of different circles that can be drawn through a place 3 points of this point is an examination
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