Our water polo team has 15 members. I want to choose a starting team consisting of 7 players, one of whom will be the goalie (the other six positions are

There are 15 people for goalie, and 1 less for each of the other positions. So it is 15 * 14 * 13 * 12 * 11 * 10 * 9.

But because the order of the other 6 guys(or girls) don't matter, you divide by 6!. So it is:

(15 * 14 * 13 * 12 * 11 * 10 * 9)/6!

32432400/720

45045

Another approach

We have 15 ways to choose a goalie.....and for each of these, we can pick any 6 of the other 14 remaining people....so......the total  ways of choosing a team  =   

15 * C(14,6)   = 15 * 3003 =   45,045 ways