In a 52 card deck, if three cards are dealt, what is the probability that two are the same suit and one is not?

You have 3 ways that you can win this game. If the first card dealt is suit A, and we call any card of a different suit B, then the three winning draws are:

AAB
ABA
ABB

It doesn't matter what the first card dealt is, only what the 2nd and 3rd are. I'll help you with the first of these combinations. Given the first card is suit A, the probability of the second card being the same suit is 12/51, and the probability of the third card being a different suit is 39/50. So the probability of order AAB is (12*39)/(51*50). Now, can you figure out the probability of drawing cards in the order ABA and ABB? Then add those three probabilities together to get your answer.

You can check your answer by considering that this is the same as 1 minus the probability that either (a) all three cards are the same or (b) all three are different.