How many five of a kinds are possible with wildcards?

Answer

In order to get a five of a kind the game being played must have at least one wildcard. With one wildcard a player needs four of the same rank plus the wildcard. If two wildcards are used the player may also have three of the same rank plus both wildcards. So how many five of a kinds are possible?

Let's start with a single wildcard case. In this situation the only way to get five of a kind is to get four of the same rank plus the wildcard. Since you need all the suits of a rank, there are only 13 ways to get four of that rank. Adding a wildcard doesn't change this count here, so with only a single wildcard there are 13 ways to get five of a kind.

A more common case is a game with two wildcards. First we start with the 13 four of the same ranks, but now we have two wildcards, so there are 13 x 2 = 26 ways to form five of a kind. Second we consider the case where we have three of the same rank and two wildcards. We can have any three of the same rank plus the two wildcards, that is (4c3) * 13 = 52 combinations. Adding those together we get 26 + 52 = 78 five of a kinds.