Question

A team of 22 soccer players needs to choose 3 players to fill water bottles. How many different groups can be made?

Answer 1

1540

Explanation:

To figure out the answer you must compute 22 choose 3. To compute this, we can plug the values into the Combination formula, which goes like this:

`(n!)/(r!(n-r)!)`, where n is the sample size, and r is the amount being chosen. In this case, n is equal to the total 22 players, and r is the 3 players being chosen to fill the bottles. Plugging the values in, we have the answer as

`(22!)/(3!(22-3)!)`. 22 minus 3 simplifies to 19 factorial, and we can expand 22 factorial out from the numerator.

`(22(21)(20)(19!))/(3!(19!))`

We can get rid of the 19 factorial from both the numerator and the denominator, and we are left with

`(22(21)(20))/(3(2))`

We can cancel out the 3 with the 3 in 21, and we can cancel out the 2 with the 2 in 20. We are left with

`22(7)(10)`

7 times 10 is equal to 70, and 70 times 22 is equal to 1540 total combinations.