How many five-card hands chosen from a standard deck contain two hearts and three spades?

To determine the number of five-card hands chosen from a standard deck that contain two hearts and three spades, you can follow these steps:

Step 1: Calculate the number of ways to choose two hearts.
In a standard deck, there are 13 hearts. To choose two hearts, you can use the combination formula: C(n, k) = n! / (k! * (n-k)!).
C(13, 2) = 13! / (2! * (13-2)!) = 78.

Step 2: Calculate the number of ways to choose three spades.
In a standard deck, there are 13 spades. To choose three spades, you can again use the combination formula: C(n, k) = n! / (k! * (n-k)!).
C(13, 3) = 13! / (3! * (13-3)!) = 286.

Step 3: Multiply the results from Step 1 and Step 2.
To determine the total number of five-card hands with two hearts and three spades, you need to multiply the number of ways to choose two hearts with the number of ways to choose three spades.
78 * 286 = 22,308.

Therefore, there are 22,308 five-card hands chosen from a standard deck that contain two hearts and three spades.

To determine the number of five-card hands chosen from a standard deck that contain two hearts and three spades, we can break it down into steps.

Step 1: Calculate the number of ways to choose two hearts from the 13 hearts in the deck.
The number of ways to choose two hearts from 13 is given by the combination formula:
C(13, 2) = 13! / (2!(13 - 2)!) = 78

Step 2: Calculate the number of ways to choose three spades from the 13 spades in the deck.
Using the combination formula again:
C(13, 3) = 13! / (3!(13 - 3)!) = 286

Step 3: Multiply the results of Step 1 and Step 2 together to get the total number of ways to choose two hearts and three spades.
78 * 286 = 22308

Therefore, there are 22,308 five-card hands chosen from a standard deck that contain two hearts and three spades.

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