[Buzz]

Bigpooch - I follow your reasoning (and agree) until you get to B.

For A3XX suited or double-suited to the ace, could be all in same suit, but X is not deuce, which seems to be what you calculated, I get a total of only 8634 hands as follows:
AAA3s---12---4*3
AA3s3t---6---6*1
AA3n3s---24---6*2*2
AA3sXs---120---6*2*10
AA3sXt---120---6*2*10
AA3sXn---240---6*2*20
AA3nXs---240---6*2*20
A3n3n3s---12---4*3
A3n3sXs---120---4*3*10
A3n3sXn---360---4*3*30
A3n3nXs---120---4*3*10
A3sXsXs---180---4*1*10*9/2
A3sXsXn---1200---4*1*10*30
A3sXnXn---1740---4*1*30*29/2
A3nXsXs---540---4*3*10*9/2
A3nXsXn---3600---4*3*10*30
total---8634

Of the above hands, I'd probably not play the 12 A333 hands, but I'd probably play the rest of them, at least some of the time.

I'm never sure my math is correct, and there are lots of places I could have made a mistake in the above table - so I’m hardly one to suggest your math is off, but perhaps you have some duplications in your figure of 9684.

Well, okay, I can check my math. Here goes.
A3subf---8634
AAA3n---4---4*1
AA3n3n---6---6*1
A3n3n3n---4---4*1
AA3nXn---240---6*2*20
A3n3nXn---360---4*3*30
A3nXnXn---5220---4*3*30*29/2
AAAA---1
AAAX---160---4*40
AAXX---4680---6*40*39/2
AXXX---39520---4*40*39*38/6
XXXX---91390---C(40,4) 40*39*38*37/24
3333---1
333X---160
33XX---4680
3XXX---39520
total---194580
should be C(48,4) = 194580. Checks. (no deuces)

Buzz