Question

A chamber of commerce board has seven total members, drawn from a pool of twenty candidates. First, a president, secretary, and treasurer are chosen. After that, four members are chosen “at large.” How many possible boards could be chosen?

• A. 10,465,200
• B. 390,700,800
• C. 2,713,200
• D. 16,279,200
• E. 5,426,400

Hint:

When roles matter, use permutations; when roles don’t matter, use combinations.

Answer 1

The Correct Option is D

Step 1: Choose officers.
President, secretary, treasurer chosen from 20: \[ P(20,3) = 20 \times 19 \times 18 = 6840. \] Step 2: Choose at-large members.
Remaining 17 candidates, choose 4: \[ \binom{17}{4} = 2380. \] Step 3: Multiply.
\[ 6840 \times 2380 = 16,279,200. \] Final Answer: \[ \boxed{16,279,200} \]