From a group of 10 men & 5 women. 4 persons are to be selected such that 4men or 4 women in the group. Find the different number of ways.

Answer

To solve the problem of selecting 4 persons from a group of 10 men and 5 women such that all selected persons are either men or women, we can break down the solution into the following steps:

### Step-by-Step Solution:

1. **Identify the Groups**: We have two groups: 10 men and 5 women. We need to select 4 persons from these groups.

2. **Calculate the Ways to Select 4 Men**:
- We can use the combination formula \( nCr \) to find the number of ways to choose 4 men from 10.
- The combination formula is given by:
\[ nCr = \frac{n!}{r!(n-r)!} \]
- For selecting 4 men from 10, we have:
\[ 10C4 = \frac{10!}{4!(10-4)!} = \frac{10!}{4! \cdot 6!} \]
- Simplifying this:
\[ 10C4 = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = \frac{5040}{24} = 210 \]

3. **Calculate the Ways to Select 4 Women**:
- Now, we calculate the number of ways to choose 4 women from 5.
- For selecting 4 women from 5, we have:
\[ 5C4 = \frac{5!}{4!(5-4)!} = \frac{5!}{4! \cdot 1!} \]
- Simplifying this:
\[ 5C4 = \frac{5}{1} = 5 \]

4. **Combine the Results**:
- Since we can either select 4 men or 4 women, we add the two results together:
\[ \text{Total ways} = 10C4 + 5C4 = 210 + 5 = 215 \]

5. **Final Answer**: The total number of different ways to select 4 persons such that all are either men or women is **215**.