Question

A box contains six red, three green, and four blue balls. In how many ways is it possible to select a sample of four balls, without replacement, in which at most three are red?

Answer 1

You have to pick four balls, and *at most* three are red, then I would consider the case of selecting more than three - i.e. four red balls - as the number of cases you *don't* want, and subtract this from the number of total cases available.

(# ways to select 4 balls without replacement in which at most 3 are red )
= (# ways to pick any 4 balls) - (# ways to pick 4 red balls)
= 13 x 12 x 11 x 10 - 4 x 3 x 2 x 1
=17136