A nonstandard deck has $15$ different card values and $6$ suits. A joker card is added as a wild card to the deck. The rule is that in any hand, the Joker must be interpreted in a way that will give the hand its highest possible rank (i.e. $4$ of a kind, flush, three of a kind, etc.). How many possible full house hands including the Joker or not are there in the nonstandard deck?

No idea on how to start or do this problem. I understand how combinations/permutations work but no idea what to do for this problem

  • 2
    $\begingroup$ What have you tried? Can you, say, count the Full Houses that do not involve the Joker? Note: you should easily be able to look up the ways people count poker hands in a standard deck...the same methods apply here. $\endgroup$ – lulu Sep 6 '18 at 0:11

If the hand doesn't include a joker then you would need $3$ cards of the same value and another two from same value (different than the one from the $3$ cards). To the wanted number choose $2$ numerical values from the $15$ and then choose $3$ (and $2$) from the suits.

If the hand include a joker then for a full house the rest of the cards must be in two pairs of same valeud cards. As above choose $2$ numerical values and the choose $2$ from the suits for each value.

Can you perform the calculation on your own?


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