## Precalculus (6th Edition) Blitzer

The club is choosing 3 officers from a group of 10 members. And the order in which the officers are chosen matters because the president, vice president, and secretary-treasurer each have different responsibilities. Therefore, the number of permutations of 10 things taken 3 at a time needs to be found. By using the formula, ${}_{n}{{P}_{r}}=\frac{n!}{\left( n-r \right)!}$ Where $n=10,r=3$, \begin{align} & {}_{10}{{P}_{3}}=\frac{10!}{\left( 10-3 \right)!} \\ & =\frac{10!}{7!} \end{align} Solving further, we get: \begin{align} & _{10}{{P}_{3}}=\frac{10\times 9\times 8\times 7!}{7!} \\ & =10\times 9\times 8 \\ & =720 \end{align} Thus, the number of different ways in which the offices can be filled is 720.