select the correct answer.\na science club has 16 members. how many ways can a president, a vice president…

Explanation:

Step1: Identify the problem type

This is a permutation problem because the order of selection (president, vice - president, treasurer) matters. The formula for permutations of (n) objects taken (r) at a time is (P(n,r)=\frac{n!}{(n - r)!}), where (n = 16) (total number of members) and (r=3) (number of positions: president, vice - president, treasurer).

Step2: Apply the permutation formula

First, calculate (n!=16!), ((n - r)!=(16 - 3)!=13!). Then (P(16,3)=\frac{16!}{13!}). Since (n!=n\times(n - 1)\times\cdots\times(n - r+1)\times(n - r)!), we can simplify (\frac{16!}{13!}=16\times15\times14).

Calculate (16\times15 = 240), then (240\times14=3360). Wait, but let's check again. Wait, maybe I made a mistake. Wait, the number of ways to choose a president: 16 choices. Then for vice - president: 15 remaining choices. Then for treasurer: 14 remaining choices. So the total number of ways is (16\times15\times14=3360).

Wait, but let's check the options. Option D is 3360. So where was the mistake earlier? Let's re - calculate:

(16\times15 = 240), (240\times14=3360). So the correct answer should be D.

Answer:

D. 3,360