Firstly, we pick which 2 suits we work with: 4 C 2
Secondly, there are three scenarios to consider:
Scenario 1: We pick 1 card from one suit and 3 from the other: 13 C 1 * 13 C 3
Scenario 2: We pick 2 cards from one suit and 2 from the other: 13 C 2 * 13 C 2
Scenario 3: We pick 3 cards from one suit and 3 from the other: $13 C 3 * 13 C 1$
(For instance, if we picked the suits to be diamonds and hearts, then from scenario one we pick 1 from diamonds and 3 from hearts and in scenario 3 we pick 3 from diamonds and 1 from hearts).
So the final answer is $$4 C 2* (( 13 C 1 * 13 C 3 )+( 13 C 2 * 13 C 2 )+( 13 C 3 * 13 C 1 ))$$
Is this the right way of approaching this problem?