A team of 10 plyers is formed out of 22 players , if 6 particular players are always included and 4 particular players are always excluded then the number of ways in which the team can be formed, is-

Options

A. `.^(22)C_(10)`
B. `.^(18)C_(3)`
C. `.^(12)C_(4)`
D. `.^(12)C_(8)`

Answer

To solve the problem step by step, we can follow these instructions:

**Step 1: Understand the total players and the conditions.**
We have a total of 22 players. Out of these, 6 players are always included in the team, and 4 players are always excluded.

**Step 2: Calculate the remaining players available for selection.**
Since 6 players are included, we subtract these from the total:
\[ 22 - 6 = 16 \]
This means we have 16 players left after including the 6 players.

**Step 3: Exclude the 4 players who cannot be selected.**
Now, we must exclude the 4 particular players from these 16 remaining players:
\[ 16 - 4 = 12 \]
So, we are left with 12 players from which we can select additional players.

**Step 4: Determine how many more players we need to select.**
Since we need a total of 10 players and we have already included 6, we need to select:
\[ 10 - 6 = 4 \]
Thus, we need to select 4 players from the 12 available players.

**Step 5: Calculate the number of ways to choose the players.**
The number of ways to choose 4 players from 12 can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Here, \( n = 12 \) and \( r = 4 \):
\[ \binom{12}{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \cdot 8!} \]

**Step 6: Simplify the combination.**
Calculating \( \binom{12}{4} \):
\[ \binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} \]
Calculating the numerator:
\[ 12 \times 11 = 132 \]
\[ 132 \times 10 = 1320 \]
\[ 1320 \times 9 = 11880 \]

Calculating the denominator:
\[ 4 \times 3 = 12 \]
\[ 12 \times 2 = 24 \]
\[ 24 \times 1 = 24 \]

Now, divide the numerator by the denominator:
\[ \frac{11880}{24} = 495 \]

Thus, the number of ways to form the team is:
\[ \binom{12}{4} = 495 \]

**Final Answer:**
The number of ways in which the team can be formed is **495**.

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