Answer
The number of different ways in which offices can be filled is 720.
Work Step by Step
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.
