Question

A team of 17 softball players needs to choose three players to refill the water cooler. What are the number of possibilities?
(1 Point)


272

680

4,080

Answer 1

To determine how many ways we can choose 3 players from a team of 17, 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, players),
- \( r \) is the number of items to choose (in this case, players to refill the cooler),
- \( ! \) denotes factorial, which is the product of all positive integers up to that number.

In this scenario, we have \( n = 17 \) and \( r = 3 \).

Substituting these values into the formula gives us:

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

This simplifies to:

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

Now, calculating the numerator:

\[
17 \times 16 \times 15 = 4080
\]

And the denominator:

\[
3 \times 2 \times 1 = 6
\]

Now, we divide the numerator by the denominator:

\[
C(17, 3) = \frac{4080}{6} = 680
\]

Thus, the number of possibilities to choose 3 players from 17 is **680**.