Wild cards and 5K vs Royal calculations- please correct!

[Easy E]

Some of this doesn't "feel" right- please straighten out the idiot:

5K with 4 deuces wild= each card rank > 8*7*6*5*4/5! times 12 ranks (can't have 5 twos) = (6720/120)*12 = 672 five of a kind

Royal flush = each suit = 9*8*7*6*5/5! time 4 suits= 15120/120 *4 = 504 royal flushes

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)?


b) With bug (joker) for 5 card hands
[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

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

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


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)

Thanks
Easy E