How many five-card hands chosen from a standard deck contain two hearts and three spades?
you are choosing 2 of the 13 hears and 3 of the 13 spades
number = C(13,2) x C(13,3)
= 78(286)
= 22308
in factorial form:
13!/(2!11!) x 13!/(3!10!)
= 13x12/2 x 13x12x11/6
= 22308
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:
1. Calculate the number of ways to choose two hearts from the thirteen hearts in the deck:
This can be done using combinations, which can be expressed as C(n, r) = n! / (r!(n-r)!).
In this case, it's C(13, 2) = 13! / (2!(13-2)!) = 78.
2. Calculate the number of ways to choose three spades from the thirteen spades in the deck:
Again, using combinations, it's C(13, 3) = 13! / (3!(13-3)!) = 286.
3. Multiply the result obtained in step 1 by the result obtained in step 2 to determine the total number of ways to choose two hearts and 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.