Thirteen boys and eight girls go on a camping trip. In how many ways can a group of six be selected to gather firewood, given the following conditions?
(a) The group consists of two girls and four boys.

I got 20020 doing random things, but part b is messing me up!

(b) The group contains at least two girls.
ways

Question

Thirteen boys and eight girls go on a camping trip. In how many ways can a group of six be selected to gather firewood, given the following conditions?
(a) The group consists of two girls and four boys.
  
I got 20020 doing random things, but part b is messing me up!

(b) The group contains at least two girls.
  ways

queelius · Answer

First one checks out: (13 choose 4) * (8 choose 2) = 20020

queelius · Answer

Okay, the second one may be done probably in a case by case way.

(8 choose 2) * (13 choose [6 - 2]) + (8 choose 3) * (13 choose [6 - 3]) + ... + (8 choose 6) * (13 choose [6 -6])

queelius · Answer

However, there is probably a simpler way. It probably has something to do with solving the complement instead?

queelius · Answer

Btw, I typed this into Wolfram Alpha: sum (8 choose k) * (13 choose (6 - k)) k = 2 to k = 6

And got the answer: 42252