Permutations and Combinations Statistics
The Fundamental Principle of If there are Multiplication n 1 ways of doing one operation, n 2 ways of doing a second operation, n 3 ways of doing a third operation, and so forth, 2
then the sequence of k operations can be performed in n 1 n 2 n 3.. n k ways. N= n 1 n 2 n 3. n k 3
Example 1 A used car wholesaler has agents who classify cars by size (full, medium, and compact) and age (0-2 years, 2-4 years, 4-6 years, and over 6 years). Determine the number of possible automobile classifications. 4
Full(F) Medium (M) Solution 0-2 2-4 4-6 >6 0-2 2-4 4-6 >6 0-2 2-4 Compact 4-6 (C) >6 The tree diagram enumerates all possible classifications, the total number of which is 3x4= 12. 5
Example 2 Mr. X has 2 pairs of trousers, 3 shirts and 2 ties. He chooses a pair of trousers, a shirt and a tie to wear everyday. Find the maximum number of days he does not need to repeat his clothing. 6
Solution The maximum number of days he does not need to repeat his clothing is 2 3 2 = 12 7
1.2 Factorials The product of the first n consecutive integers is denoted by n! and is read as factorial n. That is n! = 1 2 3 4. (n-1) n For example, 4!=1x2x3x4=24, 7!=1 2 3 4 5 6 7=5040. Note 0! defined to be 1. 8
The product of any number of consecutive integers can be expressed as a quotient of two factorials, for example, 6 7 8 9 = 9!/5! = 9! / (9 4)! 11 12 13 14 15= 15! / 10! =15! / (15 5)! In particular, n (n 1) (n 2)... (n r + 1) 9
(A) 1.3 Permutations Permutations A permutation is an arrangement of objects. abc and bca are two different permutations. 10
1. Permutations with repetition The number of permutations of r objects, taken from n unlike objects, can be found by considering the number of ways of filling r blank spaces in order with the n given objects. If repetition is allowed, each blank space can be filled by the objects in n different ways. 11
1 2 3 4 r n n n n n Therefore, the number of permutations of r objects, taken from n unlike objects, each of which may be repeated any number of times = n n n... n(r factors) = n r 12
2. Permutations without repetition If repetition is not allowed, the number of ways of filling each blank space is one less than the preceding one. 1 2 3 4 r n n-1 n-2 n-3 n-r+1 13
Therefore, the number of permutations of r objects, taken from n unlike objects, each of which can only be used once in each permutation =n(n 1)(n 2)... (n r + 1) Various notations are used to represent the number of permutations of a set of n elements taken Statistics:Permutations r and at Combinations a time; 14
some of them are 15 ), (,, r n P P P r n n r ), (,, r n P P P r n n r ), (,, r n P P P r n n r ), (,, r n P P P r n n r ), (,, r n P P P r n n r ), (,, r n P P P r n n r ), (,, r n P P P r n n r n P r r n n n n r n r n r n n n n r n n = + = + = 1) 2)...( 1)( ( 1 2 )...3 ( 1 2 )...3 1)( 2)...( 1)( ( )! (! Since We have )! (! r n n P n r =
Example 3 How many 4-digit numbers can be made from the figures 1, 2, 3, 4, 5, 6, 7 when (a) repetitions are allowed; (b) repetition is not allowed? 16
Solution (a) Number of 4-digit numbers = 7 4 = 2401. (b) Number of 4 digit numbers =7 6 5 4 = 840. 17
Example 4 In how many ways can 10 men be arranged (a) in a row, (b) in a circle? Solution (a) Number of ways is = 3628800 P 10 10 18
Suppose we arrange the 4 letters A, B, C and D in a circular arrangement as shown. Note that the arrangements ABCD, BCDA, CDAB and DABC are not distinguishable. D A C B 19
For each circular arrangement there are 4 distinguishable arrangements on a line. If there are P circular arrangements, these yield 4P arrangements on a line, which we know is 4!. Hence P = 4! 4 = (4 1)! = 3! 20
Solution (b) The number of distinct circular arrangements of n objects is (n 1)! Hence 10 men can be arranged in a circle in 9! = 362 880 ways. 21
(B) Conditional Permutations When arranging elements in order, certain restrictions may apply. In such cases the restriction should be dealt with first.. 22
Example 5 How many even numerals between 200 and 400 can be formed by using 1, 2, 3, 4, 5 as digits (a) if any digit may be repeated; (b) if no digit may be repeated? 23
Solution (a) Number of ways of choosing the hundreds digit = 2. Number of ways of choosing the tens digit = 5. Number of ways of choosing the unit digit = 2. Number of even numerals between 200 and 400 is 2 5 2 = 20. 24
Solution (b) If the hundreds digit is 2, then the number of ways of choosing an even unit digit = 1, and the number of ways of choosing a tens digit = 3. the number of numerals formed 1 1 3 = 3. 25
If the hundreds digit is 3, then the number of ways of choosing an even. unit digit = 2, and the number of ways of choosing a tens digit = 3. number of numerals formed = 1 2 3 = 6. the number of even numerals between 200 and 400 = 3 + 6 = 9 26
Example 6 In how many ways can 7 different books be arranged on a shelf (a) if two particular books are together; 27
Solution (a) If two particular books are together, they can be considered as one book for arranging. The number of arrangement of 6 books = 6! = 720. The two particular books can be arranged in 2 ways among themselves. The number of arrangement of 7 books with two particular books 28
(b) if two particular books are separated? Solution (b) Total number of arrangement of 7 books = 7! = 5040. the number of arrangement of 7 books with 2 particular books separated = 5040-1440 = 3600. 29
(C) Permutation with Indistinguishable Elements In some sets of elements there may be certain members that are indistinguishable from each other. The example below illustrates how to find the number of permutations in this kind of situation. 30
Example 7 In how many ways can the letters of the word ISOS CELES be arranged to form a new word? Solution If each of the 9 letters of ISOSCELES were different, there would be P= 9! different possible words. 31
However, the 3 S s are indistinguishable from each other and can be permuted in 3! different ways. As a result, each of the 9! arrangements of the letters of ISOSCELES that would otherwise spell a new word will be repeated 3! times. 32
To avoid counting repetitions resulting from the 3 S s, we must divide 9! by 3!. Similarly, we must divide by 2! to avoid counting repetitions resulting from the 2 indistinguishable E s. Hence the total number of words that can be formed is 9! 3! 2! = 30240 33
If a set of n elements has k 1 indistinguishable elements of one kind, k 2 of another kind, and so on for r kinds of elements, then the number of permutations of the set of n elements is n!!! Quantitative Aptitude & 2Business k k k 1 r! 34
1.4 Combinations When a selection of objects is made with no regard being paid to order, it is referred to as a combination. Thus, ABC, ACB, BAG, BCA, CAB, CBA are different permutation, but they are the same combination of letters. 35
Suppose we wish to appoint a committee of 3 from a class of 30 students. We know that P 30 3 is the number of different ordered sets of 3 students each that may be selected from among 30 students. However, the ordering of the students on the committee has no significance, 36
so our problem is to determine the number of three-element unordered subsets that can be constructed from a set of 30 elements. Any three-element set may be ordered in 3! different ways, so P 30 3 is 3! times too large. Hence, if we divide P 30 3 by 3!,the result will be the number of unordered subsets of 30 elements taken 3 at a time. 37
This number of unordered subsets is also called the number of combinations of 30 elements taken 3 at a time, denoted by C 3 30 and C 30 3 1 30 3 = P 3! = 30! 27!3! = 4060 38
In general, each unordered r- element subset of a given n- element set (r n) is called a combination. The number of combinations of n elements taken r at a time is denoted by C n r or ncr or C(n, r). 39
A general equation relating combinations to permutations is n 1 n n! Cr = Pr = r! ( n r)! r! 40
Note: (1) C n n = C n 0 = 1 (2) C n 1 = n (3) C n n = C n n-r 41
Example8 If 167 C 90 +167 C x =168 C x then x is Solution: nc r-1 +nc r =n+1 C r Given 167 C 90 +167c x =168C x We may write 167C 91-1 + 167 C 91 =167+1 C 61 =168 C 91 X=91 42
Example9 If 20 C 3r = 20C 2r+5,find r Using nc r =nc n-r in the right side of the given equation,we find, 20 C 3 r =20 C 20-(2r+5) 3r=15-2r r=3 43
Example 10 If 100 C 98 =999 C 97 +x C 901 find x. Solution 100C 98 =999C 98 +999C 97 = 999C 901 +999C 97 X=999 44
Example11 If 13 C 6 + 2 13 C 5 +13 C 4 =15 C x,the value of x is Solution : 15C x= 13C 6 + 13 C 5 + 13 C 4 = =(13c 6+13 C 5 ) + (13 C 5 + 13 C 4) = 14 C 6 +14 C 5 =15C6 X=6 or x+6 =15 X=6 or 8 45
Example12 If n C r-1 =36,n C r =84 and n C r+1 =126 then find r Solution n-r+1 =7/3 * r r 1 84 36 3/2 (r+1)+1 =7/3 * r r=3 nc nc 126 84 r + 1 = = r nc nc 3 2 r = = 7 3 46
Example 13 How many different 5-card hands can be dealt from a deck of 52 playing cards? 47
Solution Since we are not concerned with the order in which each card is dealt, our problem concerns the number of combinations of 52 elements taken 5 at a time. The number of different hands is C 52 5= 2118760. 48
Example 14 6 points are given and no three of them are collinear. (a) How many triangles can be formed by using 3 of the given points as vertices? 49
Solution: Solution (a) Number of triangles = number of ways of selecting 3 points out of 6 = C 6 3 = 20. 50
b) How many pairs of triangles can be formed by using the 6 points as vertices? 51
Let the points be A, B, C, D, E, F. If A, B, C are selected to form a triangles, then D, E, F must form the other triangle. Similarly, if D, E, F are selected to form a triangle, then A, B, C must form the other triangle. 52
Therefore, the selections A, B, C and D, E, F give the same pair of triangles and the same applies to the other selections. Thus the number of ways of forming a pair of triangles = C 6 3 2 = 10 53
Example 15 From among 25 boys who play basketball, in how many different ways can a team of 5 players be selected if one of the players is to be designated as captain? 54
Solution A captain may be chosen from any of the 25 players. The remaining 4 players can be chosen in C 25 4 different ways. By the fundamental counting principle, the total number of different teams that can be formed is 25 C 24 4=265650. 55
(B) Conditional Combinations If a selection is to be restricted in some way, this restriction must be dealt with first. The following examples illustrate such conditional combination problems. 56
A committee of 3 men and 4 women is to be selected from 6 men and 9 women. If there is a married couple among the 15 persons, in how many ways can the committee be selected so that it contains the married 57
Solution If the committee contains the married couple, then only 2 men and 3 women are to be selected from the remaining 5 men and 8 women. The number of ways of selecting 2 men out of 5 = C 5 2 = 10. 58
The number of ways of selecting 3 women out of 8 =C 8 3 = 56. the number of ways of selecting the committee = lo 56 = 560. 59
Example 17 Find the number of ways a team of 4 can be chosen from 15 boys and 10 girls if (a) it must contain 2 boys and 2 girls, 60
Solution (a) Boys can be chosen in C 15 2 = 105 ways Girls can be chosen in C 10 2 = 45 ways. Total number of ways is 105 45 = 4725. 61
(b) it must contain at least 1 boy and 1 girl. Solution : If the team must contain at least 1 boy and 1 girl it can be formed in the following ways: (I) 1 boy and 3 girls, with C 15 1 C 10 3 = 1800 ways, (ii) 2 boys and 2 girls, with 4725 ways, (iii) 3 boys and 1 girl, with C 15 3 C 10 1 = 4550 ways. 62 the total number of teams is
Example 18 Mr..X has 12 friends and wishes to invite 6 of them to a party. Find the number of ways he may do this if (a) there is no restriction on choice, 63
Solution (a) An unrestricted choice of 6 out of 12 gives C 12 6= 924. 64
(b) two of the friends is a couple and will not attend separately, 65
B Solution If the couple attend, the remaining 4 may then be chosen from the other 10 in C 10 4 ways. If the couple does not attend, then He simply chooses 6 from the other 10 in C 10 6 ways. total number of ways is C 10 4 + C 10 6 = 420. 66
Example 19 Find the number of ways in which 30 students can be divided into three groups, each of 10 students, if the order of the groups and the arrangement of the students in a group are immaterial. 67
Solution Let the groups be denoted by A, B and C. Since the arrangement of the students in a group is immaterial, group A can be selected from the 30 students in C 30 10 ways. 68
Group B can be selected from the remaining 20 students in C 20 10 ways. There is only 1 way of forming group C from the remaining 10 students. 69
Since the order of the groups is immaterial, we have to divide the product C 30 10 C 20 10 C 10 10 by 3!, hence the total number of ways of forming the three groups is 1 3! 30 3 20 10 C C C 10 10 70
Example20 If n Pr = 604800 10 C r =120,find the value of r We Know that nc r.r P r = npr. We will use this equality to find r 10Pr =10Cr.r r =604800/120=5040=7 r=7 71
Example 21 Find the value of n and r n P r = n P r+1 and n C r = n C r-1 Solution : Given n P r = n P r+1 n r=1 (i) n C r = n C r-1 Solving i and ii r=2 and n=3 n-r = r-1 (ii) 72
Multiple choice Questions 73
1. Eleven students are participating in a race. In how many ways the first 5 prizes can be won? A) 44550 B) 55440 C) 120 D) 90 74
1. Eleven students are participating in a race. In how many ways the first 5 prizes can be won? A) 44550 B) 55440 C) 120 D) 90 75
2. There are 10 trains plying between Calcutta and Delhi. The number of ways in which a person can go from Calcutta to Delhi and return A) 99. B) 90 C) 80 D) None of these 76
2. There are 10 trains plying between Calcutta and Delhi. The number of ways in which a person can go from Calcutta to Delhi and return A) 99. B) 90 C) 80 D) None of these 77
3. 4P4 is equal to A) 1 B) 24 C) 0 D) None of these 78
3. 4P4 is equal to A) 1 B) 24 C) 0 D) None of these 79
4.In how many ways can 8 persons be seated at a round table? A) 5040 B) 4050 C) 450 D) 540 80
4.In how many ways can 8 persons be seated at a round table? A) 5040 B) 4050 C) 450 D) 540 81
5. If n P : n+1 P =3:4 then 13 12 value of n is A) 15 B) 14 C) 13 D) 12 82
5. If n P : n+1 P =3:4 then 13 12 value of n is A) 15 B) 14 C) 13 D) 12 83
6.Find r if 5Pr = 60 A) 4 B) 3 C) 6 D) 7 84
6.Find r if 5Pr = 60 A) 4 B) 3 C) 6 D) 7 85
7. In how many different ways can seven persons stand in a line for a group photograph? A) 5040 B) 720 C) 120 D) 27 86
7. In how many different ways can seven persons stand in a line for a group photograph? A) 5040 B) 720 C) 120 D) 27 87
8. If C = of n is C 18 18 n n+2 A) 0 B) 2 C) 8 D) None of above then the value 88
8. If C = of n is C 18 18 n n+2 A) 0 B) 2 C) 8 D) None of above then the value 89
9. The ways of selecting 4 letters from the word EXAMINATION is A) 136. B) 130 C) 125 D) None of these 90
9. The ways of selecting 4 letters from the word EXAMINATION is A) 136. B) 130 C) 125 D) None of these 91
10 If 5Pr = 120, then the value of r is A) 4,5 B) 2 C) 4 D) None of these 92
10 If 5Pr = 120, then the value of r is A) 4,5 B) 2 C) 4 D) None of these 93
Permutations and Combinations THE END