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Exactly one matching rank

CombinatoricsMedium

A short deck has $4$ suits and the $8$ highest ranks in each suit, so $32$ cards. A hand of $5$ cards is dealt. How many hands have exactly $2$ cards of one rank while the other $3$ cards have $3$ different ranks, all different from that one?

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Hint

Choose the repeated rank, its suits, then the other ranks and their suits.

Solution

Choose the repeated rank in $8$ ways and the suits of its $2$ cards in $(2 4) = 6$ ways.
Choose the $3$ other ranks from the remaining $7$ ranks in $(3 7) = 35$ ways.
Each of those $3$ cards can be in any of $4$ suits, giving $4^{3} = 64$ possibilities.
The total is $8 \times 6 \times 35 \times 64 = 107520$ .

Answer: $107520$

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