How many different 5-card hands with at least 3 hearts can be dealt from a standard deck of 52 cards?
Question
How many different 5-card hands with at least 3 hearts can be dealt from a standard deck of 52 cards?
Solution 1
To solve this problem, we need to consider two cases: when there are exactly 3 hearts and when there are 4 or 5 hearts.
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Exactly 3 hearts: There are 13 hearts in a deck of 52 cards. The number of ways to choose 3 hearts from 13 is given by the combination formula "13 choose 3", which is C(13,3). For the remaining 2 cards, we have 39 cards left (52 total - 13 hearts = 39 non-hearts). The number of ways to choose 2 cards from these 39 is given by "39 choose 2", which is C(39,2). So, the total number of ways to get exactly 3 hearts is C(13,3) * C(39,2).
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4 or 5 hearts: Similarly, the number of ways to get exactly 4 hearts is C(13,4) * C(39,1), and the number of ways to get exactly 5 hearts is C(13,5).
Adding these cases together gives the total number of 5-card hands with at least 3 hearts.
So, the answer is C(13,3)*C(39,2) + C(13,4)*C(39,1) + C(13,5).
You can calculate the combinations using the formula C(n, k) = n! / [k!(n-k)!], where n! denotes the factorial of n.
Solution 2
To solve this problem, we need to consider three scenarios: getting exactly 3 hearts, exactly 4 hearts, and exactly 5 hearts.
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Exactly 3 hearts: There are 13 hearts in a deck of 52 cards. The number of ways to choose 3 hearts from 13 is "13 choose 3", which is denoted as C(13,3). The remaining 2 cards must be from the 39 non-heart cards. The number of ways to choose 2 non-hearts from 39 is C(39,2). So, the total number of ways to get exactly 3 hearts is C(13,3) * C(39,2).
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Exactly 4 hearts: The number of ways to choose 4 hearts from 13 is C(13,4). The remaining 1 card must be from the 39 non-heart cards. The number of ways to choose 1 non-heart from 39 is C(39,1). So, the total number of ways to get exactly 4 hearts is C(13,4) * C(39,1).
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Exactly 5 hearts: The number of ways to choose 5 hearts from 13 is C(13,5). Since all 5 cards are hearts, we don't need to choose any non-heart cards. So, the total number of ways to get exactly 5 hearts is C(13,5).
Finally, add up the results from the three scenarios to get the total number of 5-card hands with at least 3 hearts.
Solution 3
To solve this problem, we need to consider three scenarios: getting exactly 3 hearts, exactly 4 hearts, and exactly 5 hearts.
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Exactly 3 hearts: There are 13 hearts in a deck of 52 cards. The number of ways to choose 3 hearts from 13 is "13 choose 3", which is denoted as C(13,3). The remaining 2 cards must be from the 39 non-heart cards. The number of ways to choose 2 non-hearts from 39 is C(39,2). So, the total number of ways to get exactly 3 hearts is C(13,3) * C(39,2).
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Exactly 4 hearts: The number of ways to choose 4 hearts from 13 is C(13,4). The remaining 1 card must be from the 39 non-heart cards. The number of ways to choose 1 non-heart from 39 is C(39,1). So, the total number of ways to get exactly 4 hearts is C(13,4) * C(39,1).
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Exactly 5 hearts: The number of ways to choose 5 hearts from 13 is C(13,5). Since all 5 cards are hearts, we don't need to choose any non-heart cards. So, the total number of ways to get exactly 5 hearts is C(13,5).
Finally, add up the results from the three scenarios to get the total number of 5-card hands with at least 3 hearts.
Similar Questions
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