The number of ways in which a team of eleven players can be selected from from 22 players so that 2 particular players are always included and 4 naughty boys are excluded is

Options

A. `"^16 C_9`
B. `"^16 C_11`
C. `"^20 C_9`
D. `"^20 C_5`

Answer

To solve the problem of selecting a team of 11 players from 22 players, where 2 specific players must be included and 4 specific players must be excluded, we can follow these steps:

### Step 1: Determine the total number of players available for selection
Initially, we have 22 players. However, since 2 players must always be included and 4 players must be excluded, we need to adjust the total number of players available for selection.

- Players included: 2 (let's call them A and B)
- Players excluded: 4 (let's call them C, D, E, and F)

Thus, the number of players available for selection is:
\[ 22 - 4 = 18 \text{ (after excluding C, D, E, and F)} \]

### Step 2: Calculate the number of players we still need to select
Since we have already included 2 players (A and B), we need to select:
\[ 11 - 2 = 9 \text{ players} \]

### Step 3: Determine the number of players left to choose from
After excluding the 4 naughty boys, we have:
\[ 18 - 2 = 16 \text{ players left (excluding A and B)} \]

### Step 4: Calculate the number of ways to choose the remaining players
Now, we need to choose 9 players from the remaining 16 players. The number of ways to choose \( r \) players from \( n \) players is given by the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]

In this case, we need to calculate:
\[ \binom{16}{9} \]

### Step 5: Calculate \( \binom{16}{9} \)
Using the combination formula:
\[ \binom{16}{9} = \frac{16!}{9!(16-9)!} = \frac{16!}{9! \cdot 7!} \]

### Step 6: Simplify the calculation
We can simplify \( \binom{16}{9} \) as follows:
\[ \binom{16}{9} = \frac{16 \times 15 \times 14 \times 13 \times 12 \times 11 \times 10}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \]

Calculating this gives:
\[ \binom{16}{9} = 11440 \]

### Final Answer
Thus, the number of ways in which a team of 11 players can be selected from 22 players, ensuring that 2 particular players are included and 4 naughty boys are excluded, is:
\[ \boxed{11440} \]

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