Message Board : Poker Combination Exercise

For some reason my response to this post isn’t posting, so I’ll just post it in the forums and others can follow suit as well.

The number I believe is 1784.

13 quads with 4 different suited kickers each and 4 royal flushes.

Easy peasy.

The hard part is counting the broadway straights.

There are 2 basic suit patterns for a straight where no flush is possible: xxyyz and xxyzw.

For xxyyz straights there are 16 ways to distribute the suits across them and 27 ways to distribute the ranks across each of those.  That makes 432 combinations.

For xxyzw straights there are 24 ways to distribute the suits across them and 54 ways to distribute the ranks across each of those.  That makes 1296 combinations.

So adding it up:

4 royal flushes
52 quads
432 xxyyz straights
1296 xxyzw straights
———————-
1784 nuts-on-the-board

Did I get it?