I have never played poker in my life and i have to solve this complicated problem.

How many five card poker hands contain at least $3$ jacks?

Here is what i know:

There are $52$ cards in a deck. A,2,3,4,5,6,7,8,9,10,J,Q,K;

$13$ cards and $4$ suits: diamonds, clubs, spades, and hearts.

5 poker hands:_ _ _ _ _

would be it $(52, 48)$ and then $(52, 26)$?

how would you computate it??

  • $\begingroup$ Hint: Use the formula $\frac{n!}{k!(n-k)!}$, and then add i and ii $\endgroup$
    – ammie
    Jun 15, 2013 at 23:20
  • $\begingroup$ or n!/k!(n-k)! then you will be find it haha $\endgroup$
    – ammie
    Jun 15, 2013 at 23:34

1 Answer 1


One needs to know nothing about poker. Indeed you need to know only that there are $52$ different cards in a standard deck, of which $4$ are Jacks.

We are asked how many $5$-card hands there are that contain at least $3$ Jacks.

We have at least $3$ Jacks if we have (i) exactly $3$ Jacks or (ii) exactly $4$ Jacks.

(ii) We count first the $4$-Jack hands. A hand has $5$ cards, so the number of $4$-Jack hands is the number of ways to pick the non-Jack card that will keep the four Jacks company. There are $48$ non-Jacks, so there are $48$ $4$-Jack hands. This looks a little nicer as $\binom{48}{1}$.

(i) We now count the $3$-Jack hands. Which Jacks are in the hand? They can be chosen in $\binom{4}{3}$ ways. For any choice of Jacks, the two non-Jacks can be chosen from the $48$ non-Jacks in the deck in $\binom{48}{2}$ ways. So the number of $3$-Jack hands is $\binom{4}{3}\binom{48}{2}$.

For the number of hands that have at least $3$ Jacks, add the counts obtained in (ii) and (i).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.