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

suits ranks
Poker Hand         No of Combinations        Probability--------------------------------------------------------Royal Flush                         4       0.0000015391Straight Flush                     36       0.0000138517Four of a kind                    624       0.0002400960Full House                      3,744       0.0014405762Flush                           5,108       0.0019654015Straight                       10,200       0.0039246468Three of a Kind                54,912       0.0211284514Two Pair                      123,552       0.0475390156Pair                        1,098,240       0.4225690276No Pair                     1,302,540       0.5011773940Total                       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 $\binom{52}{5}$