[BruceZ]

Now, what are the hands with 1 rank card and 4 deuces- are they royal flushes or 5K or both?
There are 20 hands that meet this criteria. Do we subtract them out of both, or just the more common hand (i.e five of a kind, so it would be 652 vs. 504)?


Subtract the 20 from the lower ranking hand, since you will count these as the higher ranking hand. Normally 5-of-a-kind is higher ranking than any straight flush including a royal flush. It depends on your rules. The only standard version of poker that allows 5-of-a-kind would be draw played with a joker, and then the joker only counts as an ace or a bug to complete straights and flushes, so you can only make 5 aces, and this is the highest hand.

[NOTE- I believe I am misusing the !factorial notation here. 7!= 7*6*5*4*3*2*1 normally. I think the proper notation should be 7!5.... and the correct uniqueness equation would be (7!5)/5!, or 21]

5K- each rank = 1 chance (5!/5!) * 13 chances = 13 five of a kind
Royal- each suit > 6!/5! * 4 = 24 royal flushes


You want C(7,5) = 7!/(5!2!) = 21. This is combinations of 7 things taken 5 at a time. P(7,5) = 7!/5!, this is permuatations of 7 things taken 5 at a time which counts order. C(7,5) = P(7,5)/2!

C(7,5)*4 = 84.

c) With 2 jokers:
5K => 6!/5! * 13 = 78 five of a kind
Royal = 7!/5! * 4 = 21* 4= 84 royals

C(6,5)*13 = 78
C(7,5)*4 = 84

d) With 3 wilds: 2 jokers, 2 of hearts wild (for easiest approximation; would depend on the card- if one-eyed jacks and suicide king are the wildcards, the calculations should change?)

5K> 7!/5! * 13 = 273
Royal> 8!/5! * 4 = 224 royals

C(7,5)*13 = 273
C(8,5)*4 = 224

Someone braver than me can try the other card rankings. I think four of a kinds would have to be calculated in this manner:

Natural, non-deuce kicker +
3 ranks, 1 wild, one kicker +
2 ranks, 2 wild, one kicker +
1 rank, 3 wild, one kicker not a straight flush kicker
= sum of one rank's 4 of a kind hand * 12 (again, using 2's as wild, four twos wouldn't be possible)

So,
Natural= 4*3*2*1*40/5!= 8 of a rank
3+1+k = 4*3*2*4*40/5! = 32 of rank
2+2+k= 4*3*4*3*40/5! = 48
1+3+k = 4*4*3*2*40/5! = 32
= 120 *12 ranks
= 1440 4 of a kinds?

(in a five card hand, there are 624 non-wild 4 of a kind hands = 48 for each rank * 13 ranks. I'm not sure why factorials don't work for that calculation- I must be thinking incorrectly about this)

There are only 44 naturals of each rank since there is only one way to pick the 4 cards, and 44 non-deuce kickers. You don't have to divide by factorials when multiplying combinations, only when converting permutations to combinations.

Natural= 44 of a rank
3+1+k = 4*4*44 = 704 of a rank
2+2+k= 6*6*44 = 1584 of a rank
1+3+k = 4*4*44 = 704 of a rank
= 3036 *12 ranks = 36,432

Or you could just compute C(8,4) minus the 1 hand with 4 deuces to get [C(8,4)-1]*44 = 69*44 = 3036 of a rank. 3036*12 = 36,432 as above.