Permutations (key words: distinct; with order).

Example 1. If 4 paintings ( A, B, C, and D ) are entered in an art show, in how many different ways can the judge award a first prize and a second prize?

The number of permutations of r objects selected from n distinct objects is P(n,r) = n(n-1)(n-2)****(n-r+1) = n!/(n-r)!

Example 2. In how many different ways can a person choose 2 paintings from the 4 paintings in Example 1 ?

Solution. As it is assumed here that the order in which 2 paintings are selected does not matter, we have AB, AC, AD, BC, BD, and CD 6 ways. C(4,2) = P(4,2)/2! = 6.

The number of ways in which r objects can be selected from a set of n distinct objects is C(n,r) = P(n,r)/r!