I have a deck of **52 playing cards** and I play a game where I need a combination of specific card categories to make combos.

For example: I draw **5 cards as a sample hand** and I need exactly **1 Jack** and **1 Heart** to make a combo.

I can find the posibility of opening 1 random Jack and 1 random Heart by using the Multivariate Hypergeometic Distribution, by using these numbers:

* Hand Size n: 5
* Deck Size M: 52
* Success Jack: 1
* Possible Jacks: 4
* Success Hearts: 1
* Possible Hearts: 13

the result being 0,0020% (very low because I only want 1 of each).

The problem is that the Jack of Hearts is a possible success in both the Jack- and the Heart category, so he will appear as a success twice in the combinations.

1. What can I do to make sure this doesn't happen?
2. Is it possible to count the Jack of Hearts as a success in both categories, so I don't need an additional Heart or Jack?

//EDIT//

I think I get the hang of it, but need to test it out in practice.

I still draw **5 cards** as a sample hand from a **52 deck of cards**. But now I change the criteria, so I need

* 1 Heart (of 13 possible)
* 1 Jack (of 4 possible)
* 1 King or Queen (of 8 possible)
* 1 Clubs or Spade (of 26 possible)
* "+ 1 other"

As far as I can see, this would result in 24 succesful outcomes (4!. Correct me if I'm wrong). I'm not gonna write them all here, just wanna know if I use the right approach

One of the successful outcomes would consist of the following combination:

**1 Heart (not Jack, Queen or King)**, **1 Jack of Clubs or Spades**, **1 Queen of Heart** and **1 King or Queen of Clubs or Spades**.

$10\times 2\times 1\times 4 \times {52-10-1-31 \choose 1}=800$ (the -31 would be because they are all overlapping events at this point, except the Diamonds from A-10)

When I have calculated all 24 outcomes the probability would be calculated like this: $\frac{800+outcome2+outcome3+.....outcome'n}{2598960}$

Would this be correct?