Counting flushes

From a standard 52-card deck, how many distinct 5-card hands are flushes in the strict poker sense — all one suit but not a straight flush? (Subtract the straight flushes from the all-one-suit count.)

Show hints (2)+
  1. All one-suit hands =4×(135)=5148= 4\times\binom{13}{5} = 5148; now remove the straight flushes.
  2. Straight flushes =4= 4 suits ×10\times 10 runs (ace low and high) =40= 40; subtract.

Answer

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5108

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Asked at: SIG, Akuna Capital

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