Question

1. how many 5 - card hands can have at least 3 hearts in your hand?

Explanation:

Step1: Calculate the number of ways to get 3 hearts and 2 non - hearts

The number of hearts in a deck is 13, and the number of non - hearts is \(52 - 13=39\).
The number of ways to choose 3 hearts out of 13 is \(C(13,3)=\frac{13!}{3!(13 - 3)!}=\frac{13\times12\times11}{3\times2\times1}=286\).
The number of ways to choose 2 non - hearts out of 39 is \(C(39,2)=\frac{39!}{2!(39 - 2)!}=\frac{39\times38}{2\times1}=741\).
By the multiplication principle, the number of 5 - card hands with 3 hearts and 2 non - hearts is \(C(13,3)\times C(39,2)=286\times741 = 212926\).

Step2: Calculate the number of ways to get 4 hearts and 1 non - heart

The number of ways to choose 4 hearts out of 13 is \(C(13,4)=\frac{13!}{4!(13 - 4)!}=\frac{13\times12\times11\times10}{4\times3\times2\times1}=715\).
The number of ways to choose 1 non - heart out of 39 is \(C(39,1)=\frac{39!}{1!(39 - 1)!}=39\).
By the multiplication principle, the number of 5 - card hands with 4 hearts and 1 non - heart is \(C(13,4)\times C(39,1)=715\times39=27885\).

Step3: Calculate the number of ways to get 5 hearts

The number of ways to choose 5 hearts out of 13 is \(C(13,5)=\frac{13!}{5!(13 - 5)!}=\frac{13\times12\times11\times10\times9}{5\times4\times3\times2\times1}=1287\).

Step4: Sum up the three cases

The number of 5 - card hands with at least 3 hearts is the sum of the number of hands with 3 hearts, 4 hearts, and 5 hearts.
\(N = C(13,3)\times C(39,2)+C(13,4)\times C(39,1)+C(13,5)\)
\(N=212926 + 27885+1287=242098\)

Answer:

\(242098\)