In how many ways can a team of 5 players be selected from 8 players so as not to include a particular player?

Options

A. 42
B. 35
C. 21
D. 20

Answer

To solve the problem of selecting a team of 5 players from 8 players while excluding a particular player, we can follow these steps:

### Step 1: Identify the number of players available for selection
Since we need to exclude a particular player from our selection, we are left with 7 players (8 total players - 1 excluded player).

### Step 2: Determine the number of players to select
We need to form a team of 5 players from the remaining 7 players.

### Step 3: Use the combination formula
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 our case, we need to calculate \( \binom{7}{5} \).

### Step 4: Apply the combination formula
Substituting \( n = 7 \) and \( r = 5 \):
\[ \binom{7}{5} = \frac{7!}{5!(7-5)!} = \frac{7!}{5! \cdot 2!} \]

### Step 5: Simplify the factorials
Calculating the factorials:
\[ \binom{7}{5} = \frac{7 \times 6 \times 5!}{5! \times 2 \times 1} = \frac{7 \times 6}{2 \times 1} \]

### Step 6: Calculate the result
Now, simplifying the expression:
\[ \frac{7 \times 6}{2 \times 1} = \frac{42}{2} = 21 \]

### Final Answer
Thus, the number of ways to select a team of 5 players from 8 players, excluding a particular player, is **21**.

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