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?

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.

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mathjohndough mathjohndough · Answer 2 · 2018-11-23T14:28:20+01:00

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 [tex]P(n,r)=\frac{n!}{(n-r)!} \\P(8,3)=\frac{8!}{(8-3)!} =\frac{8\times7\times6\times5!}{5!} =8\times7\times6=336[/tex]

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 [tex]p=8\times7\times6=336[/tex]. Both ways of working this problem out are valid.