How many five-card hands chosen from a standard deck contain two hearts and three spades?
How do I solve this?
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To solve this problem, you can use the concept of combinations. Here's how you can find the number of five-card hands chosen from a standard deck that contain two hearts and three spades:
1. Determine the total number of ways to choose two hearts from the 13 hearts in a deck: This can be calculated using combinations, denoted as "C(n, k)". In this case, there are 13 hearts to choose from, and you want to select two of them, so the calculation would be C(13, 2).
2. Determine the total number of ways to choose three spades from the 13 spades in a deck: Again, use combinations to calculate how many ways you can select three spades from the 13 available. This would be C(13, 3).
3. Multiply the two results from steps 1 and 2: Since selecting two hearts can be done in C(13, 2) ways and selecting three spades can be done in C(13, 3) ways, the total number of five-card hands with two hearts and three spades is C(13, 2) * C(13, 3).
4. Calculate the result: Use the combination formula C(n, k) = n! / (k! * (n - k)!), where n! represents the factorial of n. Substitute the values and calculate the final answer.
Therefore, to solve this problem, you would multiply the result of C(13, 2) by C(13, 3) to find the total number of five-card hands chosen from a standard deck that contain two hearts and three spades.