# Game of Cards

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13 decks of cards have been mixed. What is the minimum number of cards that must be taken out from the above-mixed cards to guarantee at least one ‘four of a kind?

SherlockHolmes Expert Asked on 13th November 2022 in

To guarantee at least one “four of a kind,” we need to find the worst-case scenario in which we keep taking cards out without getting a four of a kind. In other words, we want to keep removing the cards that are least likely to help us get a four of a kind.

There are 13 ranks in a standard deck of cards (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King), so there are 13 possible four of a kinds. In order to guarantee getting at least one four of a kind, we need to have at least 13 * 4 = 52 cards in our hand (since we need one card of each rank to make a four of a kind).

However, the question asks for the minimum number of cards that must be taken out from the mixed cards to guarantee a four of a kind. So we need to think about the worst-case scenario, where we keep taking out cards without getting a four of a kind.

The worst-case scenario occurs when we take out 48 cards without getting a four of a kind. At this point, we have 4 cards from each rank in our hand, and the next card we draw will complete at least one four of a kind. So the minimum number of cards that must be taken out to guarantee at least one four of a kind is 48.

Moshe Expert Answered on 21st February 2023.

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