The following calculates probabilities for standard
5-card poker hands.

suits ranks

Poker Hand No of Combinations Probability
--------------------------------------------------------
Royal Flush 4 0.0000015391
Straight Flush 36 0.0000138517
Four of a kind 624 0.0002400960
Full House 3,744 0.0014405762
Flush 5,108 0.0019654015
Straight 10,200 0.0039246468
Three of a Kind 54,912 0.0211284514
Two Pair 123,552 0.0475390156
Pair 1,098,240 0.4225690276
No Pair 1,302,540 0.5011773940

Total 2,598,960 1.0000000000
Calculations:
Royal Flush\binom{4}{1}
Straight Flush\binom{10}{1}\binom{4}{1}-\binom{4}{1}
Four of a kind\binom{13}{1}\binom{4}{4}\binom{12}{1}\binom{4}{1}
Full House\binom{13}{1}\binom{4}{3}\binom{12}{1}\binom{4}{2}
Flush\binom{13}{5}\binom{4}{1}-\binom{10}{1}\binom{4}{1}
Straight\binom{10}{1}\binom{4}{1}^5-\binom{10}{1}\binom{4}{1}
Three of a Kind\binom{13}{1}\binom{4}{3}\binom{12}{2}\binom{4}{1}^2
Two Pair\binom{13}{2}\binom{4}{2}^2\binom{11}{1}\binom{4}{1}
Pair\binom{13}{1}\binom{4}{2}\binom{12}{3}\binom{4}{1}^3
No Pair\left [ \binom{13}{5}-10 \right ]\left [ \binom{4}{1}^5-4 \right ]
Total\binom{52}{5}