The gym has eleven different types of machines in the weight room. Geoff has time to use only three of them this afternoon. How many different combinations of machines can Geoff choose from to use?

Here the key word is combination, i. e. the order of using the machines is not important, it's the choice of the three machines.

One way to calculate:

first machine: 11 choices

second machine : 10 choices

third machine: 9 choices

Total number of ways in order: 11*10*9

But there are 6=3! ways the three machines could be ordered

so the combination is actually 11*10*9 / (3*2*1) = 165

Another way:

The number of combinations of r choices taken from n objects is

n! / (r! (n-r) !)

=11! / (3! (8!))

=165

as before.