Combinatorics question

Mar 2019
1
0
maryland
I have a standard deck of 52 cards. How do I find the number of hands of 13 cards that contain 4 cards of the same rank? (A,2-10, J,Q,K)

I'm starting off with (48 choose 9) but I am unsure of how to account for the multiples that have been calculated (the Inclusion, Exclusion principles)



Very stuck on this, please help!
 

romsek

MHF Helper
Nov 2013
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California
Do you want exactly 1 quad or at least 1 quad?
 

Plato

MHF Helper
Aug 2006
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I have a standard deck of 52 cards. How do I find the number of hands of 13 cards that contain 4 cards of the same rank? (A,2-10, J,Q,K)
I'm starting off with (48 choose 9) but I am unsure of how to account for the multiples that have been calculated (the Inclusion, Exclusion principles)
Since you did not answer, we assume you are asking for the number of thirteen card hands which has at least one foursome of the same rank.

You should note that there can be one, two or three possible foursomes in a hand. So let's use inclusion/exclusion for those three.
$\displaystyle \sum\limits_{k = 1}^3 {(-1)^{k+1}\dbinom{13}{k}\cdot\dbinom{52 - 4k}{13 - 4k}}=21717689136 $ SEE HERE.
 
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