Counting Subsets with Specific Element Requirements

[Gammage]

Homework Statement

How many subsets S \subseteq {1,2,...,21} are there if S is required to contain 5 odd integers and 6 even integers?


2. The attempt at a solution
I am having trouble breaking this one down. If the subsets contain 5 odd and 6 even, do they only contain 5 odd and 6 even? That would be 11 elements in the set. So the first element would have 11/21 chance of being odd, the second would have 10/20,... until 7/17 for the fifth. The sixth would have a 10/16 chance of being even, seventh a 9/15,...and the eleventh would have 5/11. Am I even going the right direction?

 

[e(ho0n3]

In how many ways can you pick 5 odd integers? In how many ways can you pick 6 even integers?

 

[Gammage]

(\stackrel{11}{5}) odd and (\stackrel{10}{6}) even?

 

[e(ho0n3]

Correct. And together?

 

[Gammage]

Thanks! I understand it now.