There are fifteen players for a cricket match
In how many ways the 11 players can be selected including a particular player?

Answer

To solve the problem of selecting 11 players from a total of 15 players, including a particular player, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to select 11 players from a total of 15 players, with the condition that one specific player must be included in the selection.

2. **Fix the Particular Player**: Since one player is already chosen (the particular player), we only need to select the remaining players. This means we need to select 10 more players from the remaining 14 players.

3. **Use the Combination Formula**: The number of ways to choose \( r \) players from \( n \) players is given by the combination formula \( \binom{n}{r} \). In our case, we need to calculate \( \binom{14}{10} \).

4. **Apply the Combination Identity**: We can use the identity \( \binom{n}{r} = \binom{n}{n-r} \) to simplify our calculation. Thus, \( \binom{14}{10} = \binom{14}{4} \).

5. **Calculate \( \binom{14}{4} \)**:
\[ \binom{14}{4} = \frac{14!}{4!(14-4)!} = \frac{14!}{4! \cdot 10!} \]
This simplifies to:
\[ \binom{14}{4} = \frac{14 \times 13 \times 12 \times 11}{4 \times 3 \times 2 \times 1} \]

6. **Perform the Calculation**:
- Calculate the numerator:
\[ 14 \times 13 = 182 \]
\[ 182 \times 12 = 2184 \]
\[ 2184 \times 11 = 24024 \]
- Calculate the denominator:
\[ 4 \times 3 \times 2 \times 1 = 24 \]
- Now divide the numerator by the denominator:
\[ \frac{24024}{24} = 1001 \]

7. **Final Answer**: Therefore, the number of ways to select 11 players including a particular player is **1001**.