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2 What is the probability that a five-card poker hand contains two pairs (that is, two of each of two different ranks and a fifth card of a third rank)? My attempt: Let us first pick the 3 different ranks. There are ${13choose 3}$ ways of doing this. Out of each rank consisting of 4 suits, we must pick 2 cards, 2 cards and 1 card respectively. So, no. of ways $={13choose 3}cdot {4choose 2}cdot {4choose 2}cdot {4choose 1}$ Total no. of ways of selecting a five-card poker hand $={52choose 5}$ $p=dfrac{{13choose 3}cdot {4choose 2}cdot {4choose 2}cdot {4choose 1}}{{52choose 5}}$ This doesn't match the answer given in the textbook. Where have I gone wrong? probability share | cite | improve this question