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}