How many committee of five person with a chairperson can be selected from `12` persons ?

Options

A. `960`
B. `3960`
C. `2960`
D. `1960`

Explanation

To solve the problem of how many committees of five persons with a chairperson can be selected from 12 persons, we can follow these steps:

### Step-by-Step Solution:

1. **Select the Chairperson**:
We need to select 1 chairperson from the 12 persons. The number of ways to choose 1 chairperson from 12 persons is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of persons and \( r \) is the number of persons to choose.

\[ \text{Ways to choose chairperson} = \binom{12}{1} = 12 \]

2. **Select the Other Members**:
After selecting the chairperson, we need to select 4 more members from the remaining 11 persons (since one person has already been chosen as the chairperson). The number of ways to choose 4 members from 11 persons is:

\[ \text{Ways to choose 4 members} = \binom{11}{4} \]

3. **Calculate \( \binom{11}{4} \)**:
Using the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \):

\[ \binom{11}{4} = \frac{11!}{4!(11-4)!} = \frac{11!}{4! \cdot 7!} \]

This simplifies to:

\[ = \frac{11 \times 10 \times 9 \times 8}{4 \times 3 \times 2 \times 1} \]

Calculating this step-by-step:

- The numerator: \( 11 \times 10 = 110 \)
- \( 110 \times 9 = 990 \)
- \( 990 \times 8 = 7920 \)

- The denominator: \( 4 \times 3 = 12 \)
- \( 12 \times 2 = 24 \)
- \( 24 \times 1 = 24 \)

Now divide the numerator by the denominator:

\[ \frac{7920}{24} = 330 \]

4. **Total Committees**:
Now, we multiply the number of ways to choose the chairperson by the number of ways to choose the other members:

\[ \text{Total ways} = \text{Ways to choose chairperson} \times \text{Ways to choose 4 members} = 12 \times 330 = 3960 \]

### Final Answer:
The total number of committees of five persons with a chairperson that can be selected from 12 persons is **3960**.

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