Question

A chamber of commerce board has seven total members, drawn from a pool of twenty candidates. There are two stages in the board's election process. First, a president, secretary, and treasurer are chosen. After that, four members are chosen to be "at large" without any specific title or district. How many possible boards could be chosen?

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

Hint:

When selecting people for multiple roles, calculate the number of ways to select each group and multiply the results.

Answer 1

The Correct Option is B

Step 1: Number of ways to choose the president, secretary, and treasurer.
We are selecting 3 people from 20 candidates: \[ \text{Ways to choose president, secretary, and treasurer} = \binom{20}{3} = 20 \times 19 \times 18 = 6840 \]
Step 2: Number of ways to choose the remaining 4 members.
After selecting the first 3, there are 17 candidates left, and we need to select 4 more members: \[ \text{Ways to choose 4 members} = \binom{17}{4} = \frac{17 \times 16 \times 15 \times 14}{4 \times 3 \times 2 \times 1} = 2380 \]
Step 3: Total number of possible boards.
Multiplying the two results: \[ \text{Total ways} = 6840 \times 2380 = 16,279,200 \]
Final Answer: \[ \boxed{16,279,200} \]