Bridge Hands: 5/6/2 Card Combination

[Raerin]

A bridge hand consists of 13 cards. How many bridge hands include 5 cards of one suit, 6 cards of a second suit and 2 cards of a third suit?

 

[MarkFL]

What if the question asked instead:

How many bridge hands include 5 cards of hearts, 6 cards of spades and 2 cards of diamonds?

Wold you be able answer that?

 

[Raerin]

What if the question asked instead:

How many bridge hands include 5 cards of hearts, 6 cards of spades and 2 cards of diamonds?

Wold you be able answer that?

13C5 * 13C6 * 13C2 = 172,262,376

If my question is the same as this one then my textbook's answer key is wrong. The textbook says the answer is 4 xxx, xxx, xxx

 

[MarkFL]

13C5 * 13C6 * 13C2?

If my question is the same as this one then my textbook's answer key is wrong.

Yes, good! :D That is correct, but this is for one specific combination of suits only.

Now you want to make it general. You want to multiply this by the number of ways to choose 3 suits from 4.

 

[MarkFL]

Yes, good! :D That is correct, but this is for one specific combination of suits only.

Now you want to make it general. You want to multiply this by the number of ways to choose 3 suits from 4.

I realized after I left that we need to find the permutations, not the combinations regarding the four suits, since order matters in this case because there are a different number of each suit. Hence, the number $N$ of the described bridge hands is:

\(\displaystyle N=\frac{4!}{(4-3)!}\cdot{13 \choose 5}\cdot{13 \choose 6}\cdot{13 \choose 2}=4134297024\)