Respuesta :
There are 2,607,532 5-card hands with at least two cards with the same rank.
There are 52 cards in a standard deck, and there are 4 of each rank (Ace, 2, 3, ..., King). To find the number of 5-card hands with at least two cards with the same rank, we can first find the number of 5-card hands without any pair and then subtract this from the total number of 5-card hands. The total number of 5-card hands is simply the number of ways to choose 5 cards from a deck of 52, which is 52C5 = 2, 628, 032.
To find the number of 5-card hands without a pair, we can use the principle of inclusion-exclusion. First, we choose one of the 13 ranks for the first card. Then, we choose one of the 12 remaining ranks for the second card, one of the 11 remaining ranks for the third card, one of the 10 remaining ranks for the fourth card, and one of the 9 remaining ranks for the fifth card. This gives us a total of 13 * 12 * 11 * 10 * 9 = 22, 040 ways to choose 5 cards without a pair. However, this counts the 5-card hands with 3 or more of a single rank as having no pair. To correct for this, we need to add back in the 5-card hands with 3 of a single rank and subtract the 5-card hands with 4 of a single rank.
To count the 5-card hands with 3 of a single rank, we choose one of the 13 ranks for the first 3 cards, then choose one of the 12 remaining ranks for the fourth card, and one of the 11 remaining ranks for the fifth card. This gives us a total of 13 * 12 * 11 = 1, 596 ways to choose 5 cards with 3 of a single rank.
To count the 5-card hands with 4 of a single rank, we choose one of the 13 ranks for the first 4 cards, and one of the 12 remaining ranks for the fifth card. This gives us a total of 13 * 12 = 156 ways to choose 5 cards with 4 of a single rank.
Thus, the total number of 5-card hands without a pair is 22, 040 - 1, 596 + 156 = 20, 500. The total number of 5-card hands with at least two cards with the same rank is then 2628,032 - 20,500 = 2,607,532. Therefore, there are 2,607,532 5-card hands with at least two cards with the same rank.
To learn more about principle of inclusion-exclusion,
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