Question

A team of 17 softball players need to choose three players to refill the water cooler. How many different ways are there to select 3 players?

Answer 1

Answer: 680 different combinations

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Explanation:

If order mattered, then we'd have 17*16*15 = 4080 different permutations. Notice how I started with 17 and counted down 1 at a a time until I had 3 slots to fill. We count down by 1 because each time we pick someone, we can't pick them again.

So we have 4080 different ways to pick 3 people if order mattered. But again order doesn't matter. All that counts is the group itself rather than the individual or how they rank. There are 3*2*1 = 6 ways to order any group of three people, which means there are 4080/6 = 680 different combinations possible.

An alternative is to use the nCr formula with n = 17 and r = 3. That formula is

$_n C _r = \frac{n!}{r!*(n-r)!}$

where the exclamation marks indicate factorials