The number of 13-card hands that contain four aces would be C(48,9)
= 48!/(9! 39!) = 1,677,106,640.
The number of 13-card hands that contain four kings would also be 1,677,106,640, but some of the 4-king hands were already counted, as they also contain 4 aces. The number of 13-card hands containing BOTH 4 aces AND 4 kings would be C(44,5) =...
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The number of 13-card hands that contain four aces would be C(48,9)
= 48!/(9! 39!) = 1,677,106,640.
The number of 13-card hands that contain four kings would also be 1,677,106,640, but some of the 4-king hands were already counted, as they also contain 4 aces. The number of 13-card hands containing BOTH 4 aces AND 4 kings would be C(44,5) = 1,086,008.
The number of 13-card hands that contain 4 queens is 1,677,106,640, but 1,086,008 of those hands also contain four aces; and 1,086,008 of the 4-queen hands also contain four kings; on the other hand, there are 10 hands that contain 4 aces AND 4 kings AND 4 queens...and we don't want to subtract these hands more than once.
So the answer to your question will be
13*1,677,106,640 - C(13,2)*1,086,008 +
+ C(13,3)*10
= 13*1,677,106,640 - 78*1,086,008 + 2860.
Now use a calculator to simplify that last line!