How to determine combinations correctly

[Mathematicize]

Hey everyone,

I have a question regarding combinations and poker! Could someone explain to me why the number of distinct ways to receive a full house from a five card draw is:

​

13 nCr 1 * 4 nCr 3 * 12 nCr 1 * 4 nCr 2
and not,

​

13 nCr 2 * 4 nCr 3 * 4 nCr 2

I usually make small mistakes here and there with combinatorics and I can never find a good answer from anyone. Also, I am more interested in the not in this case. Intuitively they seem to say the same thing to me. (choose two distinct denominations, choose 3 cards for the first denomination and 2 cards for the second denomination). Also does anyone have a good way at viewing combinations other than the number of subsets of size r from a set of size n?

Thanks!

 

[chiro]

Hey Mathematicize and welcome to the forums.

It might help the readers here if you show your thought process on how you derived your answer because doing this will help us understand where the thinking was wrong or even if the answer that has been provided is wrong if this is the case.

 

[Mathematicize]

The first answer is correct after computing the probabilities and checking with various sources. So I will explain my thought process on the second answer in which I obtained. First I did 13 nCr 2 because in a full house we need 2 distinct denominations where order does not matter. Next, we must pick the suit for the first denomination, and there are 4 suits and 3 that we must choose in which order does not matter, so I did 4 nCr 3. Next I needed 2 suits for the second denomination. So with the same reasoning as my last step I get 4 nCr 2. Then since each step is independent of each other, I invoke the multiplication rule to get the total number of ways to receive a full house.

The different between the answers is off by a factor of 1/2. I have a hard time visualizing this or any real way at checking my work for correctness which is a big problem.

 

[Stephen Tashi]

(choose two distinct denominations, choose 3 cards for the first denomination and 2 cards for the second denomination).

You have to account for the number of ways that one of the denominations can be designated as "the first" deonomination. When you enumerated picking the denominations, you didn't enumerate it as picking a "first" and "second" denomination.

 

[blue_raver22]

Hello,

Thank you for your question regarding combinations and poker. The reason why the number of distinct ways to receive a full house from a five card draw is 13 nCr 1 * 4 nCr 3 * 12 nCr 1 * 4 nCr 2 and not 13 nCr 2 * 4 nCr 3 * 4 nCr 2 is because the first combination represents choosing one denomination for the three cards and another denomination for the two cards. This follows the rule of nCr (n choose r) where n represents the total number of options and r represents the number of choices.

In the second combination, choosing 13 nCr 2 means choosing two denominations for the three cards, which is not possible since a full house can only have two different denominations. Additionally, the combination 4 nCr 2 represents choosing two cards from the same denomination, which would not result in a full house.

One way to view combinations is to think of it as choosing a subset of objects from a larger set. In the case of poker, you are choosing a subset of cards from a deck of 52 cards. The number of combinations is determined by the number of options for each choice and the number of choices in total.

I hope this helps clarify your understanding of combinations and their application in poker. Let me know if you have any further questions.