### Instruction: Our club has 25 members, and wishes to pick a president, secretary, and treasurer.  In how many ways can we choose the officers, if individual members can only hold at most one office?### Response: I need to count the number of possible arrangements of three different people out of 25, where the order matters.
This is a permutation problem, not a combination problem, because the roles of president, secretary, and treasurer are distinct.
To find the number of permutations of n objects taken r at a time, I can use the formula nPr = n! / (n-r)!.
In this case, n is 25, the total number of club members, and r is 3, the number of offices to fill.
So the number of ways to choose the officers is 25P3 = 25! / (25-3)!.
To simplify this expression, I can cancel out some of the factorial terms.
I get 25! / 22! = (25 x 24 x 23 x 22!) / 22! = 25 x 24 x 23.
Multiplying these numbers, I get 25 x 24 x 23 = 13,800.
So there are 13,800 ways to choose the officers of the club.