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

Answer

To solve the problem of selecting 11 players from a total of 15 players for a cricket match, we can use the concept of combinations. Here’s a step-by-step solution:

### Step 1: Identify the total number of players and the number of players to be selected
We have a total of \( n = 15 \) players, and we need to select \( r = 11 \) players.

### Step 2: Use the combination formula
The number of ways to choose \( r \) items from \( n \) items is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
In our case, we need to calculate \( \binom{15}{11} \).

### Step 3: Substitute the values into the formula
Substituting \( n = 15 \) and \( r = 11 \):
\[ \binom{15}{11} = \frac{15!}{11!(15-11)!} = \frac{15!}{11! \cdot 4!} \]

### Step 4: Simplify the factorials
We can simplify \( 15! \) as follows:
\[ 15! = 15 \times 14 \times 13 \times 12 \times 11! \]
Now substituting this back into our equation:
\[ \binom{15}{11} = \frac{15 \times 14 \times 13 \times 12 \times 11!}{11! \cdot 4!} \]
The \( 11! \) in the numerator and denominator cancels out:
\[ \binom{15}{11} = \frac{15 \times 14 \times 13 \times 12}{4!} \]

### Step 5: Calculate \( 4! \)
Now, calculate \( 4! \):
\[ 4! = 4 \times 3 \times 2 \times 1 = 24 \]

### Step 6: Substitute \( 4! \) back into the equation
Now we substitute \( 4! \) into our combination formula:
\[ \binom{15}{11} = \frac{15 \times 14 \times 13 \times 12}{24} \]

### Step 7: Perform the multiplication in the numerator
Calculating the numerator:
\[ 15 \times 14 = 210 \]
\[ 210 \times 13 = 2730 \]
\[ 2730 \times 12 = 32760 \]

### Step 8: Divide by \( 24 \)
Now, divide the result by \( 24 \):
\[ \frac{32760}{24} = 1365 \]

### Final Answer
Thus, the number of ways to select 11 players from 15 is \( \boxed{1365} \).

---