From a standard 52 -card deck, how many ways are there to pick a hand of k cards that inc

Emanuel Keith

Emanuel Keith

Answered question

2022-06-24

From a standard 52-card deck, how many ways are there to pick a hand of k cards that includes one card from all four suits?
I know that for any specific k, it's possible to break it up into cases based on the partitions of k into 4 parts. For example, if I want to choose a hand of six cards, I can break it up into two cases based on whether there are ( 1 ) three cards from one suit and one card from each of the other three or ( 2 ) two cards from each of two suits and one card from each of the other two.
Is there a simpler, more general solution that doesn't require splitting the problem into many different cases?

Answer & Explanation

Layla Love

Layla Love

Beginner2022-06-25Added 29 answers

Count the number of hands that do not contain at least one card from every suit and subtract from the total number of k-card hands. To count the number of hands that do not contain at least one card from every suit, use inclusion-exclusion considering what suits are not in a given hand. That is, letting N ( ) mean the number of hands meeting the given criteria,
N ( n o   ) + N ( n o   ) + N ( n o   ) + N ( n o   ) N ( n o   ) N ( n o   ) N ( n o   ) N ( n o   ) N ( n o   ) N ( n o   ) + N ( n o   ) + N ( n o   ) + N ( n o   ) + N ( n o   ) N ( n o   ) = 4 ( 39 k ) 6 ( 26 k ) + 4 ( 13 k ) ( 0 k ) .
So, the number of hands of k cards that include at least one card from every suit is
( 52 k ) 4 ( 39 k ) + 6 ( 26 k ) 4 ( 13 k ) + ( 0 k ) .

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