A committee of 5 members is to be formed out of 4 men ad 5 women. In how many ways this can be done.

Answer

To solve the problem of forming a committee of 5 members from 4 men and 5 women, we will break it down into cases based on the number of men in the committee.

### Step-by-Step Solution:

1. **Identify the Total Members**: We have 4 men and 5 women, making a total of 9 individuals.

2. **Define Cases Based on Men Count**:
- Case 1: 0 men and 5 women
- Case 2: 1 man and 4 women
- Case 3: 2 men and 3 women
- Case 4: 3 men and 2 women
- Case 5: 4 men and 1 woman

3. **Calculate Each Case Using Combinations**:
- **Case 1**:
- Number of ways = \( \binom{4}{0} \times \binom{5}{5} = 1 \times 1 = 1 \)
- **Case 2**:
- Number of ways = \( \binom{4}{1} \times \binom{5}{4} = 4 \times 5 = 20 \)
- **Case 3**:
- Number of ways = \( \binom{4}{2} \times \binom{5}{3} = 6 \times 10 = 60 \)
- **Case 4**:
- Number of ways = \( \binom{4}{3} \times \binom{5}{2} = 4 \times 10 = 40 \)
- **Case 5**:
- Number of ways = \( \binom{4}{4} \times \binom{5}{1} = 1 \times 5 = 5 \)

4. **Sum All Cases**:
- Total ways = \( 1 + 20 + 60 + 40 + 5 = 126 \)

5. **Final Answer**:
- Therefore, the total number of ways to form the committee is **126**.