A committee of 5 persons is to be formed from a group of 6 gentleman and 4 ladies. In how many ways can this be done if the committee is to include at least one lady ?

Answer

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

### Step-by-Step Solution:

1. **Understanding the Problem:**
We need to form a committee of 5 persons from 6 gentlemen and 4 ladies, with the condition that there is at least one lady in the committee.

2. **Total Combinations Without Restrictions:**
First, we calculate the total number of ways to form a committee of 5 persons from 10 people (6 gentlemen + 4 ladies) without any restrictions:
\[ \text{Total ways} = \binom{10}{5} \]
\[ \binom{10}{5} = \frac{10!}{5! \cdot 5!} = 252 \]

3. **Calculating the Cases with At Least One Lady:**
We will consider the cases based on the number of ladies in the committee.

- **Case 1: 1 Lady and 4 Gentlemen**
- Choose 1 lady from 4: \(\binom{4}{1}\)
- Choose 4 gentlemen from 6: \(\binom{6}{4}\)
\[ \text{Ways} = \binom{4}{1} \cdot \binom{6}{4} = 4 \cdot 15 = 60 \]

- **Case 2: 2 Ladies and 3 Gentlemen**
- Choose 2 ladies from 4: \(\binom{4}{2}\)
- Choose 3 gentlemen from 6: \(\binom{6}{3}\)
\[ \text{Ways} = \binom{4}{2} \cdot \binom{6}{3} = 6 \cdot 20 = 120 \]

- **Case 3: 3 Ladies and 2 Gentlemen**
- Choose 3 ladies from 4: \(\binom{4}{3}\)
- Choose 2 gentlemen from 6: \(\binom{6}{2}\)
\[ \text{Ways} = \binom{4}{3} \cdot \binom{6}{2} = 4 \cdot 15 = 60 \]

- **Case 4: 4 Ladies and 1 Gentleman**
- Choose 4 ladies from 4: \(\binom{4}{4}\)
- Choose 1 gentleman from 6: \(\binom{6}{1}\)
\[ \text{Ways} = \binom{4}{4} \cdot \binom{6}{1} = 1 \cdot 6 = 6 \]

4. **Total Combinations with At Least One Lady:**
Now, we add the number of ways from all cases:
\[ \text{Total ways with at least one lady} = 60 + 120 + 60 + 6 = 246 \]

### Final Answer:
The total number of ways to form a committee of 5 persons that includes at least one lady is **246**.