I'm looking at this question, and I know the answer. You have 10 blue and 12 red marbles, select two at random, what's the probability that you get at least one blue? I know I can find the probability of obtaining all red marbles, $\displaystyle \frac{12\cdot11}{22\cdot 21}$ and then find one minus this to get 5/7.

But I initially tried doing this by finding total number of ways of obtaining one blue marble (10) plus total number of ways of obtaining two blue marbles (10C2) divided by total number of ways of selecting any two marbles (22C2), and my answer is off by a factor of 3. Can anyone explain what's going wrong in this calculation?