President, secretary and treasurer

Text

In a club of $10$ members a president, a secretary and a treasurer must be chosen (distinct offices; one person cannot hold two). In how many ways is this possible?

Solution

The offices are different, so order matters and there are no repetitions: these are simple arrangements of $10$ elements taken $3$ at a time. The president is chosen in $10$ ways, the secretary among the remaining $9$, the treasurer among the remaining $8$: $N = 10\cdot 9\cdot 8 = \frac{10!}{7!} = \boxed{720}.$