What is the probability that at least 2 are pasta dinners?

Assume that there are 16 frozen dinners: 8 pasta, 4 chicken, and 4 seafood dinners. The student selects 5 of them.

What is the probability that at least 2 of the dinners selected are pasta dinners?

I thought the answer would be this:
(c(8,2)c(8,3)+c(8,3)c(8,2)+c(8,4)c(8,1))/c(16,5)
= 3696/4368, but this is incorrect. What am I doing wrong? :?

I think I figured it out:

First, I found Pr[that none selected are pasta] = (8*7*6*5*4)/(16*15*14*13*12) = 1/78

Next step: I found Pr[that at least one selected is pasta] = (c(8,1)c(8,4))/c(16,5) = 5/39

I then subracted 1 - 1/78 - 5/39 = .8589

Did I do this correctly? What if I wanted to find the probability of at at least two that were pasta and at least two that were seafood? What would I do then??