Dealt foru cards, how many hands consit of 2 pairs? i have the answer

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In summary, it was found that 2,808 four card hands consist of two pairs. This was found by solving for the sum of (13 choose 2) * (4 choose 2)^2.
  • #1
mr_coffee
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Hello everyone. I was loooking over some notes and I noticed this problem i had a question mark next too.

Supose that you are dealt four cards instead of 5. How many 4 card hands consist of two pairs (two cards of one denomination and two cards of a second denomoniation)?

The answer is 2,808.

It was found by:
(13 choose 2) * (4 choose 2)^2

But I'm confused because if you are choosing 2 cards of one denomination, that's like choosing two 5's. There is only one 5 in every 13 cards or suites correct?

so if you have
(13 choose 2) that means your choosing 2 cards out of 1 suit weither it be hearts, diamonds, clubs or spades. So how can you be choosing 2 of the same numbers from a set of 13?

Or are they saying, they are chooosing 2 cards of 1 suit, such as a 4 and a 10, and later they are choosing 2 cards to match that suit, such as a 10H and a 10C, that would be one (4 choose 2) and the other would be 4H, and a 4D that would be the other (4 choose 2)?

I think i got it by writing this question but just making sure, thanks!
 
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  • #2
Let nCr mean "n choose r".

It's 13C2 because we need to choose 2 numbers that will be the two doubles, and there are only 13 numbers to choose from. Hence, we choose 2 from the 13 numbers.

Now, for the first number we have chosen, we must choose 2 suits out of four. That's your standard deck, so we have 4C2. Similarly for the second card we have chosen for the other double, so we have 4C2 again.

Hence, (13C2)(4C2)(4C2) = (13C2)(4C2)^2.
 
  • #3
Ahh I get it now, thanks again!
 
  • #4
mr_coffee said:
Ahh I get it now, thanks again!

I'm glad you got it.

Now, I'm having a hard time with my Probability assignment myself. :frown:
 
  • #5
Post it up! hah i doubt I could help though, at least i'll be ready for my stat classes coming up from all this probability in descrete math, wee.
 
  • #6
Yeah, I should be ready for Stats next term though.

Not sure how far your Probability goes, but it does go into complicated territory.

I will be posting up questions soon. :smile:
 

1. What is the probability of being dealt a hand with 2 pairs?

The probability of being dealt a hand with 2 pairs is approximately 23.5%, or 1 in 4.25 hands.

2. How many possible combinations of 2 pairs can be made with 4 cards?

There are 6 possible combinations of 2 pairs that can be made with 4 cards.

3. What is the expected number of hands with 2 pairs in a deck of 52 cards?

On average, there will be approximately 123,552 hands with 2 pairs in a deck of 52 cards.

4. How does the number of possible hands with 2 pairs change if there are 5 or 6 cards dealt?

If 5 cards are dealt, there are 12 possible combinations of 2 pairs. If 6 cards are dealt, there are 20 possible combinations of 2 pairs.

5. Can a hand have more than 2 pairs?

No, a hand can only have 2 pairs at most. If a player is dealt more than 2 pairs, they must choose which 2 pairs to keep and discard the rest of the cards.

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