Once again, the fastest way to this answer is by way of generating functions.

Find the coefficient of the expansion of .

But you seem not to be using generating functions in the class.

So here is tedious way.

is the number of ways to select a dozen including at least three of type A.

is the number of ways to select a dozen including at least three of type Aand least three of type B.

Thus,

is the number of ways to select a dozen including at least three of type A.but no more than two of type B