Permutations

BY THE SYMBOL n! ("n factorial") we mean the product of consecutive numbers 1 through n. n! = 1 x 2 x 3 x . . . . x (n - 2) x (n - 1) x n
=n x (n - 1) x (n - 2) x . . . .x 3 x 2 x 1 The order of the factors does not matter, whether backwards or forwards. 0! is defined as 1. 0! = 1 (The usefulness of this definition will become clear as we continue.) Example. 6! = 1 x 2 x 3 x 4 x 5 x 6 = 720 Any factorial less than n! is a factor of n!. Counting Principles
Counting principles describe the total number of possibilities or choices for certain selections. The two fundamental counting principles are listed below. Fundamental Counting Principle 1: If the number of events be n, and the number of outcomes for each event in an experiment be Ti (such that i = 1 for the first event, 2 for the second event, ..., and n for the nth event), then the total number of outcomes for all event is T1 x T2 x . . . . x Tn. Example: A movie theater sells 3 sizes of popcorn (small, medium, and large) with 3 choices of toppings (no butter, butter, extra butter). How many possible ways can a bag of popcorn be purchased? Answer: 3 x 3 = 9 Fundamental Counting Principle 2: If the number of mutually exclusive (no common elements) experiments be m, and the total number of outcomes for all events in each experiment be xj (such that j = 1 for the first experiment, 2 for the second experiment, ..., and m for the mth experiment), then the total number of outcomes for all experiments is x1 + x2 + ... + xm.
Arrangements Sets (n Permutations m)
Permutation is the selection of subsets from a set of elements when the order of the selected elements is a factor. BY THE PERMUTATIONS of the letters abc we mean all of their possible arrangements: abc acb bac bca cab cba There are 6 permutations of three different things The arrangements of n elements taken m at a time represent, in fact, a partial permutation. Older books use the notation nPm (n Permutation m) for arrangements: nPm = n! / (n-m)! The number of permutations of n different things taken n at a time is n!.
Important Notes: a) Number of permutations of n things, taken r at a time, when a particular thing is to be always included in each arrangement = r^n-1Pr-1 (b) Number of permutations of n things, taken r at a time, when a particular thing is fixed: = n-1 Pr-1 (c) Number of permutations of n things, taken r at a time, when a particular thing is never taken: = n-1 Pr. (d) Number of permutations of n things, taken r at a time, when m specified things always come together = m! x ( n-m+1) ! (e) Number of permutations of n things, taken all at a time, when m specified things always come together = n ! - [ m! x (n-m+1)! ] (f) The number of permutations of n elements with n1 of repeated element, n2 of another repeated element, etc. taken n at a time. P=n!/(n1!)(n2!) (g) The number of arrangement of n different things, taken r at a time, when each may be repeated any number of times in each arrangement, is n^r
Circular permutations There are two cases of circular-permutations: - (a) If clockwise and anti clock-wise orders are different, then total number of circular-permutations is given by (n-1)! (b) If clock-wise and anti-clock-wise orders are taken as not different, then total number of circular-permutations is given by (n-1)!/2! Number of circular-permutations of n different things taken r at a time:- (a) If clock-wise and anti-clockwise orders are taken as different, then total number of circular-permutations = nPr /r (b) If clock-wise and anti-clockwise orders are taken as not different, then total number of circular permutation = nPr/2r

Examples