Assistance on how to approach a problem regarding possible combinations.

Could anyone refer me to the right direction on how to solve a problem regarding playing cards? It is a 3 Card Poker game, using 12 cards that are made up four suits (Clubs, Diamonds, Hearts, Spades), and three face values ( 2, 3, 4)

The game is rigged with the first two players cheating by having both 3-of-a-kinds (e.g. 3 of hearts, diamonds, clubs) leaving the other two players with whatever is left in the deck of 12 cards which are made up of 2's, 3's, and 4's.

I've made a table of what possible hands Player 1 and Player 2 can have. I have to find all the other possible combinations that Player 3 & 4 could having regarding the fact that the first two player always have a 3-of-a-kinds in the face values of 2,3,4 in their hands.


2C 2D 2H
2C 2D 2S
2C 2H 2S
2D 2H 2S


3C 3D 3H
3C 3D 3S
3C 3H 3S
3D 3H 3S


4C 4D 4H
4C 4D 4S
4C 4H 4S
4D 4H 4S

Thanks in advance for any assistance :)
Last edited:


TS Contributor
You want to enumerate/list all possible combinations, or just to count the total numbers of combination, given two 3-of-a-kinds are distributed to two players?

Also need to confirm that you are treating card with same number but different suits are distinct.
I believe that the professor wants it given in a sample space for each possible hand that Player 3 and 4 could have. For instance, Player 1 could have a 2 of Hearts, Diamonds, and Spades, and Player 2 has 3 of Hearts, Diamonds, and Spades.

That leaves a 2 of Clubs, 3 of Clubs, and all suits of 4's for the other two players. So a possible sample space for Player 3 could be 2 of Clubs, 3 of Clubs, and 4 of Clubs, leaving the 4 of Diamonds, Spades, Hearts in the fourth player's hand.

But then I have the choice of replacing the third player's 4 of Clubs with something in the fourth player's hand has such as 4 of Diamonds. That would create another possible hand that Player 3 could have in the scenario if Player 1 still has a 2 of Hearts, Diamonds, and Spades, and Player 2 has 3 of Hearts, Diamonds, and Spades in their hands.

Yes. Cards with the same number but from a different suit would be distinct from one another. I guess one could say that they would be mutually exclusive.

I appreciate the reply, BGM :)