Problem Solving

I find it difficult sometimes to guess the intentions of questions written this way, because "two items of every type of pen" does not make sense, particularly with only two types of pen. But I gather we just want to count how many ways to pick two pens of each type, then two ballpoints and three fountains, then three ballpoints and two fountains, and so on, until we cover every situation where we pick at least two ballpoints and at least two fountain pens. I suppose if someone had memorized a formula for this type of scenario, they could skip the first two lines below, but it's a formula you'd never need on a real GMAT problem, so I'll show how it's derived:

4C2*3C2 + 4C2*3C3 + 4C3*3C2 + 4C3*3C3 + 4C4*3C2 + 4C4*3C3

= 4C2(3C2 + 3C3) + 4C3(3C2 + 3C3) + 4C4(3C2 + 3C3)

= (4C2 + 4C3 + 4C4)(3C2 + 3C3)

= 11*4

= 44