A committee of 5 persons is to be formed from a group of a 6 gentlemen and 4 ladies . In how many ways can this be done If the committee is to be included atleast one lady ?

Options

A. 123
B. 113
C. 246
D. 945

Answer

To solve the problem of forming a committee of 5 persons from a group of 6 gentlemen and 4 ladies, ensuring that there is at least one lady in the committee, we can break down the solution into several steps.

### Step-by-Step Solution:

1. **Understanding the Requirements**:
We need to form a committee of 5 persons that includes at least one lady. This means we can have different combinations of gentlemen and ladies in the committee.

2. **Possible Combinations**:
We can have the following combinations of gentlemen (G) and ladies (L):
- 1 lady and 4 gentlemen (1L, 4G)
- 2 ladies and 3 gentlemen (2L, 3G)
- 3 ladies and 2 gentlemen (3L, 2G)
- 4 ladies and 1 gentleman (4L, 1G)

3. **Calculating Each Combination**:
We will use the combination formula \( nCr = \frac{n!}{r!(n-r)!} \) to calculate the number of ways to choose the gentlemen and ladies for each case.

- **Case 1**: 1 lady and 4 gentlemen
\[ \text{Ways} = \binom{4}{1} \times \binom{6}{4} = 4 \times 15 = 60 \]

- **Case 2**: 2 ladies and 3 gentlemen
\[ \text{Ways} = \binom{4}{2} \times \binom{6}{3} = 6 \times 20 = 120 \]

- **Case 3**: 3 ladies and 2 gentlemen
\[ \text{Ways} = \binom{4}{3} \times \binom{6}{2} = 4 \times 15 = 60 \]

- **Case 4**: 4 ladies and 1 gentleman
\[ \text{Ways} = \binom{4}{4} \times \binom{6}{1} = 1 \times 6 = 6 \]

4. **Totaling the Combinations**:
Now, we add all the ways from each case to find the total number of ways to form the committee with at least one lady.
\[ \text{Total Ways} = 60 + 120 + 60 + 6 = 246 \]

5. **Final Answer**:
Therefore, the total number of ways to form the committee of 5 persons with at least one lady is **246**.