Question

Eight people audition for a choir. The choir director must choose one soprano, one altar and one soprano. In how many ways can the director fill these positions?

Answer 1

Since the the people are not auditioning for the same position, the order in which they are selected matters. This tells us that we have to calculate the permutation of 8 objects taking 3 at a time. The permutation of n objects taking r at a time is given by the formula $P(n,r)=\frac{n!}{(n-r)!} \\P(8,3)=\frac{8!}{(8-3)!} =\frac{8\times7\times6\times5!}{5!} =8\times7\times6=336$

The director can fill these positions in 366 ways.

A quicker way to solve this problem would have been to realize that there are 8 ways to fill the first position. Once the first position has been filled, there are  7 ways to fill the second position. Once the second position has been filled, there 6 ways to fill the third position. The total number of ways to fill the positions will then be $p=8\times7\times6=336$. Both ways of working this problem out are valid.

Answer 2

There are 72 different ways the director can fill the positions.

(1+2+3+4+5+6+7+8) x 2


You multiply it by 2 because there are going to be 4 other people who will not have a position, and 8/4 is 2.