Cherese started a Green Club at her school and needs to appoint a board of members. There are eight people available for these roles. How many different selections of four out of eight people could sh

Answer 1

To determine how many different selections of four people can be made from a group of eight, we can use the combination formula. The formula for combinations is given by:

\[ C(n, r) = \frac{n!}{r!(n-r)!} \]

where:

• \( n \) is the total number of items (in this case, 8),
• \( r \) is the number of items to choose (in this case, 4), and
• \( ! \) denotes factorial, the product of all positive integers up to that number.

For Cherese's situation:

\[ C(8, 4) = \frac{8!}{4!(8-4)!} = \frac{8!}{4! \cdot 4!} \]

Now we can break it down step by step:

1. Calculate \( 8! \):

   • \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \)

2. Calculate \( 4! \):

   • \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

3. Substitute into the combinations formula: \[ C(8, 4) = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} \]

4. Perform the calculations:

   • The numerator is \( 8 \times 7 = 56 \)
   • Then \( 56 \times 6 = 336 \)
   • And \( 336 \times 5 = 1680 \)

   The denominator is \( 4! = 24 \).

5. Now divide the numerator by the denominator: \[ \frac{1680}{24} = 70 \]

Thus, the number of different selections of four people out of eight is:

\[ \boxed{70} \]