A student formed a club at their school. They have 13 members, and need to elect a president, vice president, and treasurer. How many ways are there to fill these officer positions?

• A. $2197$
• B. $1716$
• C. $13$
• D. $6$

Answer: (B)

Verified step by step guidance

1. Identify the number of positions to be filled: president, vice president, and treasurer. This means there are 3 positions to fill.
2. Recognize that the order in which these positions are filled matters, as each position is distinct (president, vice president, treasurer). This is a permutation problem.
3. Determine the number of members available to fill these positions, which is 13.
4. Use the permutation formula to calculate the number of ways to arrange 13 members into 3 positions. The formula for permutations is: $P(n,r)=\frac{n!}{(n-r)!}$, where n is the total number of items to choose from, and r is the number of items to choose.
5. Substitute the values into the permutation formula: $P(13,3)=\frac{13!}{(13-3)!}$. Calculate the factorials to find the number of permutations.