In how many ways can a committee consisting of 5 men and 6 women be formed from 8 men and 10 women?

Options

A. 266
B. 5040
C. 11760
D. 86400

Answer

To solve the problem of forming a committee consisting of 5 men and 6 women from a group of 8 men and 10 women, we can break it down into steps using combinations.

### Step-by-Step Solution:

1. **Identify the number of selections needed**:
We need to select 5 men from a total of 8 men and 6 women from a total of 10 women.

2. **Use the combination formula**:
The formula for combinations is given by:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n \) is the total number of items to choose from, \( r \) is the number of items to choose, and \( ! \) denotes factorial.

3. **Calculate the number of ways to choose 5 men from 8**:
Using the combination formula:
\[ \binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{8!}{5! \cdot 3!} \]
Simplifying this:
\[ = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = \frac{336}{6} = 56 \]

4. **Calculate the number of ways to choose 6 women from 10**:
Again using the combination formula:
\[ \binom{10}{6} = \frac{10!}{6!(10-6)!} = \frac{10!}{6! \cdot 4!} \]
Simplifying this:
\[ = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = \frac{5040}{24} = 210 \]

5. **Calculate the total number of ways to form the committee**:
Since the selections of men and women are independent, we multiply the number of ways to choose men by the number of ways to choose women:
\[ \text{Total ways} = \binom{8}{5} \times \binom{10}{6} = 56 \times 210 \]
Now, calculate:
\[ 56 \times 210 = 11760 \]

### Final Answer:
The total number of ways to form the committee is **11,760**.