I am doing probability course again, and I have this problem: " If it is assumed that all $\binom {52}{5}$ poker hands are equally likely,what is the probability of being dealt one pair?". Now, my logic gives me expression: $$\frac {\binom {50}{3} \cdot 13 \cdot \binom {4}{2}} {\binom {52}{5}}$$
My logic is following: After having my pair, I can choose any 3 cards from 50 which are left. There is $\binom {4}{2}$ to choose the pair from 4 cards, and 13 types of cards to choose pair of. Now, my answer gives me $\approx 58.8$% probability of getting a pair, but my textbook suggests that the answer is $\approx 42.26$%. I don't really see any other way to look at this. Where am I wrong?