Peggy will choose 5 of her 8 friends to join her for intramural volleyball. In how many ways can she do so?

Answer

To solve the problem of how many ways Peggy can choose 5 of her 8 friends to join her for intramural volleyball, we will use the concept of combinations.

### Step-by-Step Solution:

1. **Identify the Total Number of Friends and Friends to Choose**:
- Peggy has a total of 8 friends.
- She needs to choose 5 friends.

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 this case, \( n = 8 \) and \( r = 5 \), so we need to calculate \( \binom{8}{5} \).

3. **Substitute the Values into the Formula**:
- Substitute \( n \) and \( r \) into the combination formula:
\[ \binom{8}{5} = \frac{8!}{5!(8-5)!} = \frac{8!}{5! \cdot 3!} \]

4. **Expand the Factorials**:
- We can expand \( 8! \) as follows:
\[ 8! = 8 \times 7 \times 6 \times 5! \]
- Therefore, we can rewrite the combination:
\[ \binom{8}{5} = \frac{8 \times 7 \times 6 \times 5!}{5! \cdot 3!} \]

5. **Cancel Out the Factorials**:
- The \( 5! \) in the numerator and denominator cancels out:
\[ \binom{8}{5} = \frac{8 \times 7 \times 6}{3!} \]

6. **Calculate \( 3! \)**:
- Calculate \( 3! \):
\[ 3! = 3 \times 2 \times 1 = 6 \]

7. **Substitute Back into the Equation**:
- Now substitute \( 3! \) back into the equation:
\[ \binom{8}{5} = \frac{8 \times 7 \times 6}{6} \]

8. **Simplify the Expression**:
- The \( 6 \) in the numerator and denominator cancels out:
\[ \binom{8}{5} = 8 \times 7 = 56 \]

9. **Final Answer**:
- Therefore, the total number of ways Peggy can choose 5 of her 8 friends is:
\[ \boxed{56} \]