Question

How many different three-member teams can be formed from six students?(1 point) Responses 20 20 120 120 216 216 720

Answer 1

To find the number of different three-member teams that can be formed from six students, we can use the combination formula, which is given by:

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

where:
- \( n \) is the total number of items (in this case, students),
- \( r \) is the number of items to choose (the team size),
- \( ! \) denotes factorial, which is the product of all positive integers up to that number.

For this problem:
- \( n = 6 \) (the total number of students),
- \( r = 3 \) (the number of students to choose for the team).

Now we can calculate \( C(6, 3) \):

\[
C(6, 3) = \frac{6!}{3!(6 - 3)!} = \frac{6!}{3! \cdot 3!}
\]

Calculating the factorials:
- \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \)
- \( 3! = 3 \times 2 \times 1 = 6 \)

Now substitute these values into the combination formula:

\[
C(6, 3) = \frac{720}{6 \cdot 6} = \frac{720}{36} = 20
\]

Therefore, the number of different three-member teams that can be formed from six students is \( \boxed{20} \).