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How many 13-card hands contain seven cards of one suit?

(My intuition is $\displaystyle \frac{13!}{7!6!} \cdot 4$)

I may be wrong. please correct me

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Your intuition is not correct. You have to choose a suit (${4\choose 1}$ options), then choose 7 cards from it (${13\choose 7}$ options), then choose the rest of the hand (${39\choose 6}$ options).

Together: $${4\choose 1}{13\choose 7}{39\choose 6}.$$

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