In how many ways can a cricket team of 11 players be selected out of 16 players if two particular players are to be included and one particular players is to be rejected?

Options

A. 715
B. 615
C. 915
D. 515

Answer

To solve the problem of selecting a cricket team of 11 players from a pool of 16 players, where 2 specific players must be included and 1 specific player must be excluded, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Players**:
We start with a total of 16 players.

2. **Account for Inclusion and Exclusion**:
- We have 2 players that must be included in the team.
- We have 1 player that must be excluded from the team.

3. **Calculate Available Players**:
After including the 2 players and excluding 1 player, we can calculate the number of players available for selection:
- Total players = 16
- Players included = 2
- Players excluded = 1
- Therefore, the number of players available for selection = 16 - 2 - 1 = 13 players.

4. **Determine Team Composition**:
Since 2 players are already included in the team of 11, we need to select the remaining players:
- Total players needed = 11
- Players already included = 2
- Therefore, players still needed = 11 - 2 = 9 players.

5. **Select Players from Available Pool**:
Now we need to select 9 players from the 13 available players. This is a combination problem where we use the combination formula:
\[ \text{Number of ways} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 13 \) (available players) and \( r = 9 \) (players to select).

6. **Apply the Combination Formula**:
We can substitute the values into the formula:
\[ \binom{13}{9} = \frac{13!}{9!(13-9)!} = \frac{13!}{9! \cdot 4!} \]

7. **Simplify the Factorials**:
We can simplify this expression:
\[ \binom{13}{9} = \frac{13 \times 12 \times 11 \times 10}{4 \times 3 \times 2 \times 1} \]

8. **Calculate the Result**:
- Calculate the numerator: \( 13 \times 12 = 156 \)
- \( 156 \times 11 = 1716 \)
- \( 1716 \times 10 = 17160 \)
- Calculate the denominator: \( 4! = 24 \)
- Therefore, \( \frac{17160}{24} = 715 \).

9. **Final Answer**:
Thus, the total number of ways to select the cricket team under the given conditions is **715**.