A group consists of 5 men and 5 women. If the number of different five -person committees containing k men and ( 5-k) women is 100, what is the value of k ?

Options

A. 2 only
B. 3 only
C. 2 or 3
D. 4

Answer

To solve the problem, we need to determine the value of \( k \) such that the number of different five-person committees containing \( k \) men and \( 5-k \) women is equal to 100.

### Step-by-Step Solution:

1. **Understanding the Committee Selection**:
We have a total of 5 men and 5 women. We want to form a committee of 5 people consisting of \( k \) men and \( 5 - k \) women.

2. **Using Combinations**:
The number of ways to choose \( k \) men from 5 is given by the combination formula \( \binom{5}{k} \), and the number of ways to choose \( 5 - k \) women from 5 is \( \binom{5}{5-k} \).

Therefore, the total number of different committees can be expressed as:
\[ \binom{5}{k} \times \binom{5}{5-k} \]

3. **Setting Up the Equation**:
According to the problem, this total must equal 100:
\[ \binom{5}{k} \times \binom{5}{5-k} = 100 \]

4. **Using the Property of Combinations**:
We know that \( \binom{5}{5-k} = \binom{5}{k} \). Thus, we can rewrite the equation as:
\[ \left( \binom{5}{k} \right)^2 = 100 \]

5. **Taking the Square Root**:
Taking the square root of both sides gives:
\[ \binom{5}{k} = 10 \]

6. **Finding \( k \)**:
Now we need to find \( k \) such that \( \binom{5}{k} = 10 \). We can calculate the combinations for \( k = 0, 1, 2, 3, 4, 5 \):
- \( \binom{5}{0} = 1 \)
- \( \binom{5}{1} = 5 \)
- \( \binom{5}{2} = 10 \) (This is a match)
- \( \binom{5}{3} = 10 \) (This is also a match)
- \( \binom{5}{4} = 5 \)
- \( \binom{5}{5} = 1 \)

Hence, the possible values for \( k \) are 2 and 3.

7. **Conclusion**:
Therefore, the values of \( k \) that satisfy the condition are \( k = 2 \) or \( k = 3 \).

### Final Answer:
The value of \( k \) can be either 2 or 3.

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