Business Mathematics - Permutation and Combination

 

Business Mathematics

Permutation and Combination

 

 CONTENTS:

1	Definitions
2	Formulas
3	35 Selected Problems
4	Solutions to the Selected Problems

 

Definitions

 

Permutation

A permutation is a mathematical technique that determines the number of possible arrangements in a set of items when the order of the arrangements matters. Common mathematical problems involve choosing only several items from a set of items in a certain order.

Combination

In mathematics, a combination refers to the technique that determines the number of possible selections of items / things from a set of items where the order of selections does not matter. It's a way of choosing a group of objects / things without regard to the arrangement or sequence in which they are picked.

 Formulas

 

  Formulas of Permutation

1	Permutation of ‘n’ different things taken ‘r’ at a time (nPr):
	nPr = n! / (n – r)!
2	Permutation of ‘n’ different things taken all at a time (nPn):
	nPn = nP(n – 1) = n!
3	0! = 1
4	1! = 1
5	n! = n × (n – 1) × (n – 2) ×..........× 1. [‘n!’ is read as ‘n factorial’ ]
6	nP0 = 1
7	nP1 = n
8	nPr = n × (n – 1)P(r – 1)
9	nPr = (n – 1)Pr + r × (n – 1)P(r – 1)
10	Permutation of ‘n’ things taken all at a time, where ‘n’ is represented by ‘p’ equal things, ‘q’ equal things and ‘r’ equal things
	= n! / (p! × q! × r!)
11	Permutation of ‘n’ different things each of which can be repeated ‘r’ number of times (For example, In how many ways ‘r’ letters can be posted in ‘n’ different boxes?)
	= nr
12	Number of ways in which ‘n’ persons can be seated at a round table with respect to the table
	= n!
13	Number of ways in which ‘n’ persons can be seated at a round table with respect to each other
	= (n – 1)!
14	Number of ways in which ‘n’ different stones/beads can be placed in a necklace
	= ½ × (n – 1)!
15	Permutation of ‘n’ different things taken ‘r’ at a time in which ‘p’ particular things never occur
	= (n – p)Pr
16	Permutation of ‘n’ different things taken ‘r’ at a time in which ‘p’ particular things occupy stated places
	= (n – p)P(r – p)
17	Permutation of ‘n’ different things taken ‘r’ at a time in which ‘p’ particular things are always present
	= (n – p)P(r – p) × rPp
18	Permutation of ‘n’ different things taken ‘r’ at a time in which ‘p’ particular things shall always occur together in an assigned order
	= (n – p)P(r – p) × (r – p + 1)
19	Permutation of ‘n’ different things taken all at a time in which ‘p’ particular things shall always occur in an assigned order
	= n! /p!

 

  Formulas of Combination

1	Combination of ‘n’ different things taken ‘r’ at a time (nCr):
	nCr = n! ÷ [r! × (n – r)!]
2	nCr = nPr ÷ r!
3	nCr = nC(n – r)
4	nCr = (n – 1)Cr + (n – 1)C(r – 1)
5	nC0 = 1
6	nC1 = n
7	nCn = 1
8	If nCr = nCs, either r = s, or r + s = n
9	nCr ÷ nC(r – 1) = (n – r + 1) ÷ r
10	nCr + nC(r – 1) = (n + 1)Cr
11	Number of combination of ‘n’ different things taken ‘r’ at a time where ‘p’ things will always occur
	= (n – p)C(r – p)
12	Number of combination of ‘n’ different things taken ‘r’ at a time where ‘p’ things will never occur
	= (n – p)Cr
13	(m + n)C(p + q) = mCp + nCq
14	nC1 + nC2 + nC3 +........+ nCn = 2n − 1
15	nC0 + nC1 + nC2 + nC3 +........+ nCn = 2n
16	Combination of ‘n’ things taken all together where ‘n’ is represented by ‘p’ equal things, ‘q’ equal things and ‘r’ equal things
	= (p + 1)(q + 1)(r + 1) − 1
17	Number of ways in which two groups out of (m + n) things (m ≠ n) can be formed so that one group consists of ‘m’ things and the other group consists of ‘n’ things
	= (m + n)Cm = (m + n)! ÷ (m! × n!)
18	Number of ways in which ‘m’ things can be selected from (m + n) things (m ≠ n)
	= (m + n)Cm = (m + n)! ÷ (m! × n!)
19	Number of ways in which two groups out of (m + n) things (m = n) can be formed so that each group consists of equal number of things
	= (1/2!) × [(2m)! ÷ 2(m!)]
20	Number of ways in which three groups out of (m + n + p) things (m ≠ n ≠ p) can be formed so that the groups contain ‘m’ things, ‘n’ things and ‘p’ things respectively
	= (m+n+p)Cm × (n+p)Cn × pCp
	= (m+n+p)! ÷ (m! × n! × p!)
21	Number of ways in which three groups out of (m + n + p) things (m = n = p) can be formed so that each group contains equal number of things
	= (1/3!) × [(3m)! ÷ 3(m!)]
22	Number of ways in which ‘2m’ things can be divided equally amongst 2 persons
	= (2m)! ÷ 2(m!)
23	Number of ways in which ‘3m’ things can be divided equally amongst 3 persons
	= (3m)! ÷ 3(m!)

 

CMA – Foundation

Permutation and Combination

Selected Problems

 

1. In how many ways can 5 candidates be screened for 3 vacancies for 3 different job-profiles?

 

2. In how many ways 3 cheques can be given to 4 employees when each employee is eligible to receive any number of cheques?

 

3. Find the number of arrangements which could be made with the letters of the word “APPLE”.

 

4. In how many ways 8 boys can form a ring?

 

5. In how many ways 8 different beads can be placed in a necklace?

 

6. If 1/ (4!) + 1/ (5!) = x/ (6!), find ‘x’.

 

7. There are 21 boys and 19 girls in a class. In how many ways can one boy and one girl be selected?

 

8. In how many ways the 1^st and 2^nd prizes of maths and stats can be given in a class of 40 students?

 

9. How many words can be formed by using the letters of the word “ALLAHABAD”, so that vowels will occupy even places?

 

10. There are 5 doors and 4 windows in a room. In how many ways a cat can enter and leave that room?

 

11. In how many ways 3 rows can be selected for vaccination out of 5 rows of students of a college assuming vaccination can be done only for one row at a time?

 

12. If ^{(n + 1)}P₃: ⁿP₂ = 27: 9, find ‘n’.

 

13. If 8 × ^{(n – 1)}P₃ = ^{(n – 1)}P₄, find ‘n’.

 

14. If ⁿP₅ = 42 × ⁿP₃, find ‘n’.

 

15. In how many ways 6 mobiles can be gifted to 6 employees, if no employee is eligible to receive more than one mobile?

 

16. In how many ways can 8 articles numbered from 1 to 8 be arranged, if even numbered articles are same?

 

17. In how many ways 6 books out of 10 different books can be arranged in a bookshelf so that 3 particular books are always together?

 

18. In how many ways can the letters of the word “TABLE” be arranged so that the vowels are always - (i) together, (ii) separated?

 

19. Find in how many ways the letters of the word “PURPOSE” can be rearranged

          i)        Keeping the position of the vowels fixed;

        ii)        Without changing the relative positions of the vowels and consonants.

 

20. How many numbers between 5,000 and 6,000 can be formed with the digits 3, 4, 5, 6, 7, 8 when none of the digits will be repeated in any of the numbers so formed?

21. In how many ways can the letters of the word SUNDAY be arranged? How many of them do not begin with ‘S’? How many of them do not begin with ‘S’, but end with ‘Y’?

 

22. In how many ways 5 boys and 5 girls can take their seats out of 10 seats in a round table, so that no two girls will sit side by side?

 

23. From a group of 15 men, how many selections of 9 men can be made so as to exclude 3 particular men?

 

24. There are 7 candidates for a post. In how many ways can a selection of 4 be made amongst them, so that:

   (a)        2 persons whose qualifications are below par are excluded?

   (b)        2 persons with good qualifications are included?

 

25. In how many ways can a committee of 3 ladies and 4 gentlemen be appointed from a meeting consisting of 8 ladies and 7 gentlemen? What will be the number of ways if Mrs. Sen refuses to serve in a committee having Mr. Sharma as a member?

 

26. From 7 gentlemen and 4 ladies a committee of 5 is to be formed. In how many ways can this be done to include at least one lady?

 

27. In how many ways can a boy invite one or more of his 5 friends?

 

28. In how many ways 15 things can be divided into three groups of 4, 5 and 6 things respectively?

 

29. In how many ways can the letters of the word SIGNAL be arranged so that the vowels occupy only the odd places?

 

30. How many triangles can be made by joining the vertices of a decagon? How many diagonals will it have?

 

31. There are 10 persons in a party. If one can shake hand with others, how many handshakes can be made?

 

32. How many diagonals are there in a hexagon?

 

33. There are 12 points in a plane of which 5 are collinear. Find the number of triangles that can be formed with these points.

 

34. In how many ways can a team of 11 be chosen from 14 football players two of whom can only be goalkeeper?

 

35. A polygon has 27 diagonals. Find number of its sides.