a standard deck of 52 cards contains 13 cards with hearts, 13 with diamonds, 13 with clubs, and 13 with…

Answer:

B. 6,169,176

Explanation:

Step1: Recall combination formula

The combination formula is $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n$ is the total number of items and $r$ is the number of items to be chosen.

Step2: Calculate number of ways to choose hearts

For hearts, $n = 13$ and $r=2$. So $C(13,2)=\frac{13!}{2!(13 - 2)!}=\frac{13\times12}{2\times1}=78$.

Step3: Calculate number of ways to choose diamonds

For diamonds, $n = 13$ and $r = 2$. So $C(13,2)=\frac{13!}{2!(13 - 2)!}=\frac{13\times12}{2\times1}=78$.

Step4: Calculate number of ways to choose clubs

For clubs, $n = 13$ and $r = 2$. So $C(13,2)=\frac{13!}{2!(13 - 2)!}=\frac{13\times12}{2\times1}=78$.

Step5: Calculate number of ways to choose spades

For spades, $n = 13$ and $r = 1$. So $C(13,1)=\frac{13!}{1!(13 - 1)!}=13$.

Step6: Calculate total number of 7 - card hands

By the multiplication principle, the total number of 7 - card hands is $C(13,2)\times C(13,2)\times C(13,2)\times C(13,1)=78\times78\times78\times13 = 6169176$.