A committee of 4 persons is to be formed from 2 ladies 2 old man and 4 youngmen such that it includes at least 1 lady, at least one old man and at most 2 youngmen. Then the total number of ways in which this committee can be formed is

Options

A. 40
B. 41
C. 16
D. 32

Answer

To solve the problem of forming a committee of 4 persons from 2 ladies, 2 old men, and 4 young men, with the conditions that there is at least 1 lady, at least 1 old man, and at most 2 young men, we can break it down into cases based on the number of young men included.

### Step-by-Step Solution:

1. **Identify the Variables**:
- Ladies (L): 2
- Old Men (O): 2
- Young Men (Y): 4

2. **Define Cases Based on Young Men**:
We can have two cases based on the number of young men:
- Case 1: 2 Young Men
- Case 2: 1 Young Man
- Case 3: 0 Young Men (not applicable since we need at least 1 lady and 1 old man)

3. **Case 1: 1 Lady, 1 Old Man, 2 Young Men**:
- Choose 1 lady from 2: \( \binom{2}{1} \)
- Choose 1 old man from 2: \( \binom{2}{1} \)
- Choose 2 young men from 4: \( \binom{4}{2} \)

Calculation:
\[ \text{Ways} = \binom{2}{1} \times \binom{2}{1} \times \binom{4}{2} = 2 \times 2 \times 6 = 24 \]

4. **Case 2: 1 Lady, 2 Old Men, 1 Young Man**:
- Choose 1 lady from 2: \( \binom{2}{1} \)
- Choose 2 old men from 2: \( \binom{2}{2} \)
- Choose 1 young man from 4: \( \binom{4}{1} \)

Calculation:
\[ \text{Ways} = \binom{2}{1} \times \binom{2}{2} \times \binom{4}{1} = 2 \times 1 \times 4 = 8 \]

5. **Case 3: 2 Ladies, 1 Old Man, 1 Young Man**:
- Choose 2 ladies from 2: \( \binom{2}{2} \)
- Choose 1 old man from 2: \( \binom{2}{1} \)
- Choose 1 young man from 4: \( \binom{4}{1} \)

Calculation:
\[ \text{Ways} = \binom{2}{2} \times \binom{2}{1} \times \binom{4}{1} = 1 \times 2 \times 4 = 8 \]

6. **Case 4: 2 Ladies, 2 Old Men, 0 Young Men**:
- Choose 2 ladies from 2: \( \binom{2}{2} \)
- Choose 2 old men from 2: \( \binom{2}{2} \)

Calculation:
\[ \text{Ways} = \binom{2}{2} \times \binom{2}{2} = 1 \times 1 = 1 \]

7. **Total Ways**:
Now, we sum all the cases:
\[ \text{Total} = 24 + 8 + 8 + 1 = 41 \]

### Final Answer:
The total number of ways to form the committee is **41**.