My lecture slide says the number of ways of choosing a $3$ of a kind card-hand is (assume we are finding $5$-card hands from a standard deck of cards) $ c(4,3)\times \ c(13,1) \times\ c(48,2) $. But I found a different answer in this wiki page. Could somebody please tell me which one is correct?
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1The slide formula includes full houses, wiki doesn't – WW1 Dec 07 '19 at 21:45
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Both formulae start with ${13 \choose 1}{4 \choose 3}$ for picking a kind and 3 suits. Then, your lecture slides have the factor ${48 \choose 2}$ for picking just any 2 of the 48 cards not of the chosen kind. On the other hand, the wikipedia formula has instead of this 3rd factor ${12 \choose 2}{4 \choose 1}^2$ for picking 2 cards of any combination of suits but differing in kind, for else that would score higher at poker. So it seems the latter is more exacting and accurate in its definition of "3 of a kind".
Papa Smurf
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1In particular, the answer $\binom{4}{3}\binom{13}{1}\binom{48}{2}$ allows for the possibility of a full house. – N. F. Taussig Dec 07 '19 at 21:48