3 card poker

The following explains a technique for calculating the frequency of 3-card poker hands in casinos.
The Formula is:   nCr = n! / (r! x (n - r)!)
Straight Flush:
•    There are 12 differently ranked straight flushes from A-2-3 up to Q-K-A in each of the 4 dissimilar suits.
•    Formula is 12C1 x 4C1 = 48
Three of a Kind:
•    There are 13 differently ranked three of a kind employing 3 of the 4 suits.
•    Formula is  13C1 x 4C3 = 52
Straight:
•    There are 12 differently ranked straights from A-2-3 up to Q-K-A. Each of the cards can be 1 of the 4 suits with the straight flushes being expelled.
•    Formula is  12C1 x (4C1)3 - 48 = 720
Flush:
•    A flush holds 3 of the 13 ranks, every card belonging to 1 of the 4 suits. The straight flushes are expelled.
•    Formula is  13C3 x 4C1 - 48 = 1096
One Pair:
•    There are 13 differently ranked pairs applying 2 of 4 suits, the third card being 1 of the 12 residual ranks employing 1 of the 4 suits.
•    Formula is  13C1 x 4C2 x 12C1 x 4C1 = 3744
Nothing:
•    Any hand not being one of the above kinds of hands.
•    Formula is  52C3 - 48 - 52 - 720 - 1096 - 3744 = 16440