Formula of Permutation and combinations
- Permutation : permutation is related to the act of arranging all the members of a set into some sequence or order.The number of permutation or arrangement or choice or waysof ‘r’ things from a set of n things without replacement: nPr = (n!) / (n-r)!.
Example:if n = 10 and r = 4, then nPr = (n!) / (n-r)! = 10! /(10-4)! = 10!/6! = 10 x 9 x 8 x 7 x 6! /6!
For permutation the will be given as ‘ find the number of permutation that can be made… or find the number of arrangements that can be made… or find the number of ways in which you can arrange… ‘
- Combination : combination is a way of selecting items from a collection, such that (unlike permutations) the order of selection does not matter.A combination is the choice of r things from a set of n things without replacement and where order does not matter.
For combination the problem will be showing as ‘findthe number of combination… or find the number of selection… or find the number of ways in which of select…’
Example:In how many ways a committee consisting of 5 men and 3 women, can be chosen from 9 men and 12 women?
Solution:
Choose 5 men out of 9 men = 9C5 ways =9!/5!(9 – 5)! = 9!/5!4! = 9 X 8 X 7 X 6 X5 X 4!/5x4x3x2x4! = 126 ways
Choose 3 women out of 12 women = 12C3 ways = 12!/ (12 -3)! = 12!/3!9! = 12X11X10X9!/3X2X9! = 220 ways
Total number of ways = (126 x 220)= 27720 ways
The committee can be chosen in 27720 ways.
- Number of arrangements of n items of which p, q and r are different kind of items = n!/p!q!r!.
- The number of permutations of n things, taken r at a time when each item may be repeated upto r times in any arrangements is nr.
- Out of n item selection of 1 or more itemscan be express the number of ways of selection as (2n – 1). it is also called total number of combinations.
- Number of ways of dividing (p+q) items into two groups of p and q items respectively is (p+q)!/p!q!
- The numbers of ways of dividing of 2p items into two equal groups of p each is = (2p)!/(p!)2 where two groups have distinct identity.
- The numbers of ways of dividing of 2p items into two equal groups of p each is = (2p)!/2!(p!)2 where two groups do not have distinct identity.
- The number of arrangements in circular fashion can be found by first fixing the position of one item.
- The number of circular arrangements of n distinct items is (n - 1 )!If there is a difference between clockwise and anticlockwise. And if there has no difference between clockwise and anticlockwise, then arrangements will be (n - 1 )!/2.
