From a standard 52-card deck, how many ways are there to pick a hand of...

Emanuel Keith

Emanuel Keith

Answered

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

Expert

2022-06-25Added 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 ) .

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?