7.4 Permutations

2. Permutations: What is a Permutation? A permutation of objects is an arrangement of things or objects in a definite order with no repetition. (a) For example: ART, RAT, RTA, ATR, TRA, and TAR are all different arrangements of the letters R, T, and A. Notice order is important and there is no repetition. Mathematically, the number of permutations of the 3 objects taken 3 at a time is, P3,3 = 3! = 6 In general, arranging n objects without repetition from n different objects such that order matters is, P n,n = n! = n (n – 1) (n – 2) ··(n – m + 1) where n ≥ m. (b) Now suppose we want to select a president and a vice president from a club of four. Choosing two officers from the club is a permutation of four people taken two at time. Mathematically, the number of permutations of the 4 people taken 2 at time is, 4!

P4,2 = (4 - 2)! =12 In general, arranging n objects without repetition taken r a time such that order matters is,

n!

Pn,r = (n - r)! = n (n – 1) (n – 2) ··(n – m + 1) where n ≥ m.