Of the 12 members of a high school drama club, 8 are seniors. The club plans to establish an 8 - member committee to interview potential club members. If exactly 6 members of the committee must be seniors, how many committees are possible ?

Options

A. 21
B. 34
C. 168
D. 336

Explanation

To solve the problem of forming an 8-member committee from a drama club with specific conditions on seniority, we can follow these steps:

### Step 1: Identify the total members and their categories
- Total members in the drama club = 12
- Number of seniors = 8
- Number of non-seniors = 12 - 8 = 4

### Step 2: Determine the committee composition
- The committee must consist of exactly 8 members.
- Out of these, exactly 6 members must be seniors.
- Therefore, the remaining 2 members must be non-seniors.

### Step 3: Calculate the number of ways to choose seniors
- We need to choose 6 seniors from the 8 available seniors.
- The number of ways to choose 6 seniors from 8 is given by the combination formula \( C(n, k) \), which is calculated as:
\[ C(8, 6) = \frac{8!}{6!(8-6)!} = \frac{8!}{6! \cdot 2!} = \frac{8 \times 7}{2 \times 1} = 28 \]

### Step 4: Calculate the number of ways to choose non-seniors
- We need to choose 2 non-seniors from the 4 available non-seniors.
- The number of ways to choose 2 non-seniors from 4 is:
\[ C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{4!}{2! \cdot 2!} = \frac{4 \times 3}{2 \times 1} = 6 \]

### Step 5: Calculate the total number of committees
- The total number of ways to form the committee is the product of the ways to choose seniors and non-seniors:
\[ \text{Total Committees} = C(8, 6) \times C(4, 2) = 28 \times 6 = 168 \]

### Final Answer
Thus, the total number of possible committees is **168**.

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