Question

A club with 31 members is to select five officers (president, vice president, secretary, treasurer, and historian). In how many ways can this be done?

Answer 1

Take the easy case: 2 members and you only need to select a president and a vice president: there are two ways to do this; order matters so these are permutations as opposed to combinations.

If there were 10 members:
Of the 10 members, there are 10 ways to select the president
Of the 9 remaining members,there are 9 ways to select the vice president
Of the 8 remaining members,there are 8 ways to select the sec.
Of the 7 remaining members,there are 7 ways to select the treas.
Of the 6 remaining members,there are 6 ways to select the hist.

So there are 10*9*8*7*6 = 10!/(10-5)!ways to select the officers.

Now you can apply the same reasoning to 31 members.