Question

A team of 17 softball players needs to choose three players to refill the water cooler.

The described situation is a ____ (combination, permutation)

The number of possibilities is ___.

Answer 1

The described situation is a **combination** because the order in which the players are chosen does not matter; you are simply selecting three players from a total of 17.

To calculate the number of combinations, you can use the combination formula:

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

where:
- \(n\) is the total number of players (17),
- \(r\) is the number of players to choose (3),
- \(n!\) denotes the factorial of \(n\).

Using this formula, we find:

\[
C(17, 3) = \frac{17!}{3!(17-3)!} = \frac{17!}{3! \cdot 14!}
\]

Calculating the factorials, we have:

\[
C(17, 3) = \frac{17 \times 16 \times 15}{3 \times 2 \times 1} = \frac{4080}{6} = 680
\]

Thus, the number of possibilities is **680**.

In summary:

The described situation is a **combination**. The number of possibilities is **680**.