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 group of 15, including a particular player, we can follow these steps:

### Step 1: Understand the Problem
We have 15 players, and we need to select 11 players for a cricket match. One of these players is a particular player (let's call this player P1) who must be included in the selection.

### Step 2: Fix the Particular Player
Since P1 is already included in the team, we only need to select 10 more players from the remaining players.

### Step 3: Determine the Remaining Players
After including P1, we have 14 players left (P2, P3, ..., P15) from which we need to select the remaining 10 players.

### Step 4: Use Combinations to Select the Players
The number of ways to choose 10 players from 14 can be calculated using combinations. The formula for combinations is given by:

\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

In our case, we need to calculate:

\[ \binom{14}{10} \]

### Step 5: Simplify the Combination
Using the combination formula, we have:

\[ \binom{14}{10} = \frac{14!}{10!(14-10)!} = \frac{14!}{10! \cdot 4!} \]

### Step 6: Expand the Factorials
We can simplify this further. We know that:

\[ 14! = 14 \times 13 \times 12 \times 11 \times 10! \]

Thus, we can write:

\[ \binom{14}{10} = \frac{14 \times 13 \times 12 \times 11 \times 10!}{10! \cdot 4!} \]

### Step 7: Cancel Out the Factorials
The \(10!\) in the numerator and denominator cancels out:

\[ \binom{14}{10} = \frac{14 \times 13 \times 12 \times 11}{4!} \]

### Step 8: Calculate \(4!\)
We know that:

\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 9: Substitute and Calculate
Now substituting back, we have:

\[ \binom{14}{10} = \frac{14 \times 13 \times 12 \times 11}{24} \]

Calculating the numerator:

\[ 14 \times 13 = 182 \]
\[ 182 \times 12 = 2184 \]
\[ 2184 \times 11 = 24024 \]

Now, divide by 24:

\[ \frac{24024}{24} = 1001 \]

### Final Answer
Thus, the number of ways to select 11 players including the particular player is:

\[ \boxed{1001} \]

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