How many different committes of 5 members may be formed 6 gentlemen and 4 ladies ?

Options

A. 181
B. 357
C. 603
D. 252

Answer

To solve the problem of how many different committees of 5 members can be formed from 6 gentlemen and 4 ladies, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Number of People**:
We have a total of 6 gentlemen and 4 ladies. Therefore, the total number of people is:
\[ 6 + 4 = 10 \]

2. **Determine the Size of the Committee**:
We need to form a committee of 5 members.

3. **Use the Combination Formula**:
The number of ways to choose \( r \) members from \( n \) members is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
In our case, \( n = 10 \) and \( r = 5 \).

4. **Calculate the Combinations**:
Plugging the values into the formula:
\[ \binom{10}{5} = \frac{10!}{5!(10-5)!} = \frac{10!}{5!5!} \]

5. **Simplify the Factorials**:
We can simplify \( 10! \) as follows:
\[ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5! \]
Thus, we have:
\[ \binom{10}{5} = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5!}{5! \times 5!} \]
The \( 5! \) cancels out:
\[ = \frac{10 \times 9 \times 8 \times 7 \times 6}{5!} \]

6. **Calculate \( 5! \)**:
We know that:
\[ 5! = 120 \]

7. **Perform the Multiplication**:
Now we calculate the numerator:
\[ 10 \times 9 = 90 \]
\[ 90 \times 8 = 720 \]
\[ 720 \times 7 = 5040 \]
\[ 5040 \times 6 = 30240 \]

8. **Divide by \( 5! \)**:
Now, divide the result by \( 120 \):
\[ \frac{30240}{120} = 252 \]

9. **Conclusion**:
Therefore, the total number of different committees of 5 members that can be formed from 6 gentlemen and 4 ladies is:
\[ \boxed{252} \]