Question

a standard deck of 52 cards contains 13 cards with hearts, 13 with diamonds, 13 with clubs, and 13 with spades. how many 7 - card hands are possible with 2 hearts, 2 diamonds, 2 clubs, and 1 spade? assume that the order of the cards does not matter, and use the formula for combinations to find your answer.
474,552
6,169,176
49,353,408
133,784,560

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

We want 2 hearts out of 13. Using the combination formula, $C(13,2)=\frac{13!}{2!(13 - 2)!}=\frac{13\times12}{2\times1}=78$.

Step3: Calculate number of ways to choose diamonds

We want 2 diamonds out of 13. So $C(13,2)=\frac{13!}{2!(13 - 2)!}=\frac{13\times12}{2\times1}=78$.

Step4: Calculate number of ways to choose clubs

We want 2 clubs out of 13. Thus $C(13,2)=\frac{13!}{2!(13 - 2)!}=\frac{13\times12}{2\times1}=78$.

Step5: Calculate number of ways to choose spades

We want 1 spade out of 13. So $C(13,1)=\frac{13!}{1!(13 - 1)!}=13$.

Step6: Use multiplication principle

The total number of 7 - card hands with 2 hearts, 2 diamonds, 2 clubs and 1 spade is the product of the number of ways of choosing each type of card. That is $78\times78\times78\times13$.
$78\times78\times78\times13=(78^3)\times13=(78\times78\times78)\times13=(6084\times78)\times13 = 474552\times13=6169176$.

Answer:

6,169,176