A flush in a strange five-suit deck

Question

A game designer prints an unusual deck: five suits, each with the same 10 ranks, for 50 cards total. You are dealt a random 5-card hand.

What is the probability the hand is a "flush", all five cards from the same suit?

Accepted Answer

The total number of 5-card hands from 50 cards is \binom{50}{5} = 2{,}118{,}760. To build a flush, first pick which of the 5 suits it is (5 ways), then choose 5 of that suit's 10 ranks: 5 	imes \binom{10}{5} = 5 	imes 252 = 1260. So P(	ext{flush}) = \frac{1260}{2{,}118{,}760} = \frac{63}{105{,}938} \approx 0.000595, about 1 hand in 1680. Takeaway. Hand probabilities are counts of favorable hands over \binom{n}{k}. Break a favorable hand into independent choices, which suit, then which ranks, and multiply. The same recipe scales to any deck once you swap in the right numbers.

A flush in a strange five-suit deck

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