Assume that you play a variant of a poker game with five cards that is using a subset of one ordinary deck of cards. The subset considered is cards valued 7, 8, 9,10,J,Q,K,A from the one decks, in total 32 cards. Allow A-7-8-9-10 as a low straight or straight flush. Calculate the exact probability of the following hands using the census method.

one pair two pairs three of a kind straight full house flush four of a kind straight flush nothing

Solution : Number of possible hands

Any Hand

In

In[263]:= deck = Range[7, 13]

Out[263]= {7, 8, 9, 10, 11, 12, 13}

Subsets[{7, 8, 9, 10, A, J, Q, K}, {5}]

Out[41]= {{7, 8, 9, 10, A}, {7, 8, 9, 10, J}, {7, 8, 9, 10, Q}, {7, 8, 9, 10, K}, {7, 8, 9, A, J}, {7, 8, 9, A, Q}, {7, 8, 9, A, K}, {7, 8, 9, J, Q}, {7, 8, 9, J, K}, {7, 8, 9, Q, K}, {7, 8, 10, A, J}, {7, 8, 10, A, Q}, {7, 8, 10, A, K}, {7, 8, 10, J, Q}, {7, 8, 10, J, K}, {7, 8, 10, Q, K}, {7, 8, A, J, Q}, {7, 8, A, J, K}, {7, 8, A, Q, K}, {7, 8, J, Q, K}, {7, 9, 10, A, J}, {7, 9, 10, A, Q}, {7, 9, 10, A, K}, {7, 9, 10, J, Q}, {7, 9, 10, J, K}, {7, 9, 10, Q, K}, {7, 9, A, J, Q}, {7, 9, A, J, K}, {7, 9, A, Q, K}, {7, 9, J, Q, K}, {7, 10, A, J, Q}, {7, 10, A, J, K}, {7, 10, A, Q, K}, {7, 10, J, Q, K}, {7, A, J, Q, K}, {8, 9, 10, A, J}, {8, 9, 10, A, Q}, {8, 9, 10, A, K}, {8, 9, 10, J, Q}, {8, 9, 10, J, K}, {8, 9, 10, Q, K}, {8, 9, A, J, Q}, {8, 9, A, J, K}, {8, 9, A, Q, K}, {8, 9, J, Q, K}, {8, 10, A, J, Q}, {8, 10, A, J, K}, {8, 10, A, Q, K}, {8, 10, J, Q, K}, {8, A, J, Q, K}, {9, 10, A, J, Q}, {9, 10, A, J, K}, {9, 10, A, Q, K}, {9, 10, J, Q, K}, {9, A, J, Q, K}, {10, A, J, Q, K}}

Length[Subsets[{7, 8, 9, 10, A, J, Q, K}, {5}]]

In[175]:= Subsets[{A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K}, {5}]

Out[175]= {{A, 2, 3, 4, 5}, {A, 2, 3, 4, 6}, {A, 2, 3, 4, 7}, {A, 2, 3, 4, 8}, {A, 2, 3, 4, 9}, {A, 2, 3, 4, 10}, {A, 2, 3, 4, J}, {A, 2, 3, 4, Q}, {A, 2, 3, 4, K}, {A, 2, 3, 5, 6}, {A, 2, 3, 5, 7}, {A, 2, 3, 5, 8}, {A, 2, 3, 5, 9}, {A, 2, 3, 5, 10}, {A, 2, 3, 5, J}, {A, 2, 3, 5, Q}, {A, 2, 3, 5, K}, {A, 2, 3, 6, 7}, {A, 2, 3, 6, 8}, {A, 2, 3, 6, 9}, {A, 2, 3, 6, 10}, {A, 2, 3, 6, J}, {A, 2, 3, 6, Q}, {A, 2, 3, 6, K}, {A, 2, 3, 7, 8}, {A, 2, 3, 7, 9}, {A, 2, 3, 7, 10}, {A, 2, 3, 7, J}, {A, 2, 3, 7, Q}, {A, 2, 3, 7, K}, {A, 2, 3, 8, 9}, {A, 2, 3, 8, 10}, {A, 2, 3, 8, J}, {A, 2, 3, 8, Q}, {A, 2, 3, 8, K}, {A, 2, 3, 9, 10}, {A, 2, 3, 9, J}, {A, 2, 3, 9, Q}, {A, 2, 3, 9, K}, {A, 2, 3, 10, J}, {A, 2, 3, 10, Q}, {A, 2, 3, 10, K}, {A, 2, 3, J, Q}, {A, 2, 3, J, K}, {A, 2, 3, Q, K}, {A, 2, 4, 5, 6}, {A, 2, 4, 5, 7}, {A, 2, 4, 5, 8}, {A, 2, 4, 5, 9}, {A, 2, 4, 5, 10}, {A, 2, 4, 5, J}, {A, 2, 4, 5, Q}, {A, 2, 4, 5, K}, {A, 2, 4, 6, 7}, {A, 2, 4, 6, 8}, {A, 2, 4, 6, 9}, {A, 2, 4, 6, 10}, {A, 2, 4, 6, J}, {A, 2, 4, 6, Q}, {A, 2, 4, 6, K}, {A, 2, 4, 7, 8}, {A, 2, 4, 7, 9}, {A, 2, 4, 7, 10}, {A, 2, 4, 7, J}, {A, 2, 4, 7, Q}, {A, 2, 4, 7, K}, {A, 2, 4, 8, 9}, {A, 2, 4, 8, 10}, {A, 2, 4, 8, J}, {A, 2, 4, 8, Q}, {A, 2, 4, 8, K}, {A, 2, 4, 9, 10}, {A, 2, 4, 9, J}, {A, 2, 4, 9, Q}, {A, 2, 4, 9, K}, {A, 2, 4, 10, J}, {A, 2, 4, 10, Q}, {A, 2, 4, 10, K}, {A, 2, 4, J, Q}, {A, 2, 4, J, K}, {A, 2, 4, Q, K}, {A, 2, 5, 6, 7}, {A, 2, 5, 6, 8}, {A, 2, 5, 6, 9}, {A, 2, 5, 6, 10}, {A, 2, 5, 6, J}, {A, 2, 5, 6, Q}, {A, 2, 5, 6, K}, {A, 2, 5, 7, 8}, {A, 2, 5, 7, 9}, {A, 2, 5, 7, 10}, {A, 2, 5, 7, J}, {A, 2, 5, 7, Q}, {A, 2, 5, 7, K}, {A, 2, 5, 8, 9}, {A, 2, 5, 8, 10}, {A, 2, 5, 8, J}, {A, 2, 5, 8, Q}, {A, 2, 5, 8, K}, {A, 2, 5, 9, 10}, {A, 2, 5, 9, J}, {A, 2, 5, 9, Q}, {A, 2, 5, 9, K}, {A, 2, 5, 10, J}, {A, 2, 5, 10, Q}, {A, 2, 5, 10, K}, {A, 2, 5, J, Q}, {A, 2, 5, J, K}, {A, 2, 5, Q, K}, {A, 2, 6, 7, 8}, {A, 2, 6, 7, 9}, {A, 2, 6, 7, 10}, {A, 2, 6, 7, J}, {A, 2, 6, 7, Q}, {A, 2, 6, 7, K}, {A, 2, 6, 8, 9}, {A, 2, 6, 8, 10}, {A, 2, 6, 8, J}, {A, 2, 6, 8, Q}, {A, 2, 6, 8, K}, {A, 2, 6, 9, 10}, {A, 2, 6, 9, J}, {A, 2, 6, 9, Q}, {A, 2, 6, 9, K}, {A, 2, 6, 10, J}, {A, 2, 6, 10, Q}, {A, 2, 6, 10, K}, {A, 2, 6, J, Q}, {A, 2, 6, J, K}, {A, 2, 6, Q, K}, {A, 2, 7, 8, 9}, {A, 2, 7, 8, 10}, {A, 2, 7, 8, J}, {A, 2, 7, 8, Q}, {A, 2, 7, 8, K}, {A, 2, 7, 9, 10}, {A, 2, 7, 9, J}, {A, 2, 7, 9, Q}, {A, 2, 7, 9, K}, {A, 2, 7, 10, J}, {A, 2, 7, 10, Q}, {A, 2, 7, 10, K}, {A, 2, 7, J, Q}, {A, 2, 7, J, K}, {A, 2, 7, Q, K}, {A, 2, 8, 9, 10}, {A, 2, 8, 9, J}, {A, 2, 8, 9, Q}, {A, 2, 8, 9, K}, {A, 2, 8, 10, J}, {A, 2, 8, 10, Q}, {A, 2, 8, 10, K}, {A, 2, 8, J, Q}, {A, 2, 8, J, K}, {A, 2, 8, Q, K}, {A, 2, 9, 10, J}, {A, 2, 9, 10, Q}, {A, 2, 9, 10, K}, {A, 2, 9, J, Q}, {A, 2, 9, J, K}, {A, 2, 9, Q, K}, {A, 2, 10, J, Q}, {A, 2, 10, J, K}, {A, 2, 10, Q, K}, {A, 2, J, Q, K}, {A, 3, 4, 5, 6}, {A, 3, 4, 5, 7}, {A, 3, 4, 5, 8}, {A, 3, 4, 5, 9}, {A, 3, 4, 5, 10}, {A, 3, 4, 5, J}, {A, 3, 4, 5, Q}, {A, 3, 4, 5, K}, {A, 3, 4, 6, 7}, {A, 3, 4, 6, 8}, {A, 3, 4, 6, 9}, {A, 3, 4, 6, 10}, {A, 3, 4, 6, J}, {A, 3, 4, 6, Q}, {A, 3, 4, 6, K}, {A, 3, 4, 7, 8}, {A, 3, 4, 7, 9}, {A, 3, 4, 7, 10}, {A, 3, 4, 7, J}, {A, 3, 4, 7, Q}, {A, 3, 4, 7, K}, {A, 3, 4, 8, 9}, {A, 3, 4, 8, 10}, {A, 3, 4, 8, J}, {A, 3, 4, 8, Q}, {A, 3, 4, 8, K}, {A, 3, 4, 9, 10}, {A, 3, 4, 9, J}, {A, 3, 4, 9, Q}, {A, 3, 4, 9, K}, {A, 3, 4, 10, J}, {A, 3, 4, 10, Q}, {A, 3, 4, 10, K}, {A, 3, 4, J, Q}, {A, 3, 4, J, K}, {A, 3, 4, Q, K}, {A, 3, 5, 6, 7}, {A, 3, 5, 6, 8}, {A, 3, 5, 6, 9}, {A, 3, 5, 6, 10}, {A, 3, 5, 6, J}, {A, 3, 5, 6, Q}, {A, 3, 5, 6, K}, {A, 3, 5, 7, 8}, {A, 3, 5, 7, 9}, {A, 3, 5, 7, 10}, {A, 3, 5, 7, J}, {A, 3, 5, 7, Q}, {A, 3, 5, 7, K}, {A, 3, 5, 8, 9}, {A, 3, 5, 8, 10}, {A, 3, 5, 8, J}, {A, 3, 5, 8, Q}, {A, 3, 5, 8, K}, {A, 3, 5, 9, 10}, {A, 3, 5, 9, J}, {A, 3, 5, 9, Q}, {A, 3, 5, 9, K}, {A, 3, 5, 10, J}, {A, 3, 5, 10, Q}, {A, 3, 5, 10, K}, {A, 3, 5, J, Q}, {A, 3, 5, J, K}, {A, 3, 5, Q, K}, {A, 3, 6, 7, 8}, {A, 3, 6, 7, 9}, {A, 3, 6, 7, 10}, {A, 3, 6, 7, J}, {A, 3, 6, 7, Q}, {A, 3, 6, 7, K}, {A, 3, 6, 8, 9}, {A, 3, 6, 8, 10}, {A, 3, 6, 8, J}, {A, 3, 6, 8, Q}, {A, 3, 6, 8, K}, {A, 3, 6, 9, 10}, {A, 3, 6, 9, J}, {A, 3, 6, 9, Q}, {A, 3, 6, 9, K}, {A, 3, 6, 10, J}, {A, 3, 6, 10, Q}, {A, 3, 6, 10, K}, {A, 3, 6, J, Q}, {A, 3, 6, J, K}, {A, 3, 6, Q, K}, {A, 3, 7, 8, 9}, {A, 3, 7, 8, 10}, {A, 3, 7, 8, J}, {A, 3, 7, 8, Q}, {A, 3, 7, 8, K}, {A, 3, 7, 9, 10}, {A, 3, 7, 9, J}, {A, 3, 7, 9, Q}, {A, 3, 7, 9, K}, {A, 3, 7, 10, J}, {A, 3, 7, 10, Q}, {A, 3, 7, 10, K}, {A, 3, 7, J, Q}, {A, 3, 7, J, K}, {A, 3, 7, Q, K}, {A, 3, 8, 9, 10}, {A, 3, 8, 9, J}, {A, 3, 8, 9, Q}, {A, 3, 8, 9, K}, {A, 3, 8, 10, J}, {A, 3, 8, 10, Q}, {A, 3, 8, 10, K}, {A, 3, 8, J, Q}, {A, 3, 8, J, K}, {A, 3, 8, Q, K}, {A, 3, 9, 10, J}, {A, 3, 9, 10, Q}, {A, 3, 9, 10, K}, {A, 3, 9, J, Q}, {A, 3, 9, J, K}, {A, 3, 9, Q, K}, {A, 3, 10, J, Q}, {A, 3, 10, J, K}, {A, 3, 10, Q, K}, {A, 3, J, Q, K}, {A, 4, 5, 6, 7}, {A, 4, 5, 6, 8}, {A, 4, 5, 6, 9}, {A, 4, 5, 6, 10}, {A, 4, 5, 6, J}, {A, 4, 5, 6, Q}, {A, 4, 5, 6, K}, {A, 4, 5, 7, 8}, {A, 4, 5, 7, 9}, {A, 4, 5, 7, 10}, {A, 4, 5, 7, J}, {A, 4, 5, 7, Q}, {A, 4, 5, 7, K}, {A, 4, 5, 8, 9}, {A, 4, 5, 8, 10}, {A, 4, 5, 8, J}, {A, 4, 5, 8, Q}, {A, 4, 5, 8, K}, {A, 4, 5, 9, 10}, {A, 4, 5, 9, J}, {A, 4, 5, 9, Q}, {A, 4, 5, 9, K}, {A, 4, 5, 10, J}, {A, 4, 5, 10, Q}, {A, 4, 5, 10, K}, {A, 4, 5, J, Q}, {A, 4, 5, J, K}, {A, 4, 5, Q, K}, {A, 4, 6, 7, 8}, {A, 4, 6, 7, 9}, {A, 4, 6, 7, 10}, {A, 4, 6, 7, J}, {A, 4, 6, 7, Q}, {A, 4, 6, 7, K}, {A, 4, 6, 8, 9}, {A, 4, 6, 8, 10}, {A, 4, 6, 8, J}, {A, 4, 6, 8, Q}, {A, 4, 6, 8, K}, {A, 4, 6, 9, 10}, {A, 4, 6, 9, J}, {A, 4, 6, 9, Q}, {A, 4, 6, 9, K}, {A, 4, 6, 10, J}, {A, 4, 6, 10, Q}, {A, 4, 6, 10, K}, {A, 4, 6, J, Q}, {A, 4, 6, J, K}, {A, 4, 6, Q, K}, {A, 4, 7, 8, 9}, {A, 4, 7, 8, 10}, {A, 4, 7, 8, J}, {A, 4, 7, 8, Q}, {A, 4, 7, 8, K}, {A, 4, 7, 9, 10}, {A, 4, 7, 9, J}, {A, 4, 7, 9, Q}, {A, 4, 7, 9, K}, {A, 4, 7, 10, J}, {A, 4, 7, 10, Q}, {A, 4, 7, 10, K}, {A, 4, 7, J, Q}, {A, 4, 7, J, K}, {A, 4, 7, Q, K}, {A, 4, 8, 9, 10}, {A, 4, 8, 9, J}, {A, 4, 8, 9, Q}, {A, 4, 8, 9, K}, {A, 4, 8, 10, J}, {A, 4, 8, 10, Q}, {A, 4, 8, 10, K}, {A, 4, 8, J, Q}, {A, 4, 8, J, K}, {A, 4, 8, Q, K}, {A, 4, 9, 10, J}, {A, 4, 9, 10, Q}, {A, 4, 9, 10, K}, {A, 4, 9, J, Q}, {A, 4, 9, J, K}, {A, 4, 9, Q, K}, {A, 4, 10, J, Q}, {A, 4, 10, J, K}, {A, 4, 10, Q, K}, {A, 4, J, Q, K}, {A, 5, 6, 7, 8}, {A, 5, 6, 7, 9}, {A, 5, 6, 7, 10}, {A, 5, 6, 7, J}, {A, 5, 6, 7, Q}, {A, 5, 6, 7, K}, {A, 5, 6, 8, 9}, {A, 5, 6, 8, 10}, {A, 5, 6, 8, J}, {A, 5, 6, 8, Q}, {A, 5, 6, 8, K}, {A, 5, 6, 9, 10}, {A, 5, 6, 9, J}, {A, 5, 6, 9, Q}, {A, 5, 6, 9, K}, {A, 5, 6, 10, J}, {A, 5, 6, 10, Q}, {A, 5, 6, 10, K}, {A, 5, 6, J, Q}, {A, 5, 6, J, K}, {A, 5, 6, Q, K}, {A, 5, 7, 8, 9}, {A, 5, 7, 8, 10}, {A, 5, 7, 8, J}, {A, 5, 7, 8, Q}, {A, 5, 7, 8, K}, {A, 5, 7, 9, 10}, {A, 5, 7, 9, J}, {A, 5, 7, 9, Q}, {A, 5, 7, 9, K}, {A, 5, 7, 10, J}, {A, 5, 7, 10, Q}, {A, 5, 7, 10, K}, {A, 5, 7, J, Q}, {A, 5, 7, J, K}, {A, 5, 7, Q, K}, {A, 5, 8, 9, 10}, {A, 5, 8, 9, J}, {A, 5, 8, 9, Q}, {A, 5, 8, 9, K}, {A, 5, 8, 10, J}, {A, 5, 8, 10, Q}, {A, 5, 8, 10, K}, {A, 5, 8, J, Q}, {A, 5, 8, J, K}, {A, 5, 8, Q, K}, {A, 5, 9, 10, J}, {A, 5, 9, 10, Q}, {A, 5, 9, 10, K}, {A, 5, 9, J, Q}, {A, 5, 9, J, K}, {A, 5, 9, Q, K}, {A, 5, 10, J, Q}, {A, 5, 10, J, K}, {A, 5, 10, Q, K}, {A, 5, J, Q, K}, {A, 6, 7, 8, 9}, {A, 6, 7, 8, 10}, {A, 6, 7, 8, J}, {A, 6, 7, 8, Q}, {A, 6, 7, 8, K}, {A, 6, 7, 9, 10}, {A, 6, 7, 9, J}, {A, 6, 7, 9, Q}, {A, 6, 7, 9, K}, {A, 6, 7, 10, J}, {A, 6, 7, 10, Q}, {A, 6, 7, 10, K}, {A, 6, 7, J, Q}, {A, 6, 7, J, K}, {A, 6, 7, Q, K}, {A, 6, 8, 9, 10}, {A, 6, 8, 9, J}, {A, 6, 8, 9, Q}, {A, 6, 8, 9, K}, {A, 6, 8, 10, J}, {A, 6, 8, 10, Q}, {A, 6, 8, 10, K}, {A, 6, 8, J, Q}, {A, 6, 8, J, K}, {A, 6, 8, Q, K}, {A, 6, 9, 10, J}, {A, 6, 9, 10, Q}, {A, 6, 9, 10, K}, {A, 6, 9, J, Q}, {A, 6, 9, J, K}, {A, 6, 9, Q, K}, {A, 6, 10, J, Q}, {A, 6, 10, J, K}, {A, 6, 10, Q, K}, {A, 6, J, Q, K}, {A, 7, 8, 9, 10}, {A, 7, 8, 9, J}, {A, 7, 8, 9, Q}, {A, 7, 8, 9, K}, {A, 7, 8, 10, J}, {A, 7, 8, 10, Q}, {A, 7, 8, 10, K}, {A, 7, 8, J, Q}, {A, 7, 8, J, K}, {A, 7, 8, Q, K}, {A, 7, 9, 10, J}, {A, 7, 9, 10, Q}, {A, 7, 9, 10, K}, {A, 7, 9, J, Q}, {A, 7, 9, J, K}, {A, 7, 9, Q, K}, {A, 7, 10, J, Q}, {A, 7, 10, J, K}, {A, 7, 10, Q, K}, {A, 7, J, Q, K}, {A, 8, 9, 10, J}, {A, 8, 9, 10, Q}, {A, 8, 9, 10, K}, {A, 8, 9, J, Q}, {A, 8, 9, J, K}, {A, 8, 9, Q, K}, {A, 8, 10, J, Q}, {A, 8, 10, J, K}, {A, 8, 10, Q, K}, {A, 8, J, Q, K}, {A, 9, 10, J, Q}, {A, 9, 10, J, K}, {A, 9, 10, Q, K}, {A, 9, J, Q, K}, {A, 10, J, Q, K}, {2, 3, 4, 5, 6}, {2, 3, 4, 5, 7}, {2, 3, 4, 5, 8}, {2, 3, 4, 5, 9}, {2, 3, 4, 5, 10}, {2, 3, 4, 5, J}, {2, 3, 4, 5, Q}, {2, 3, 4, 5, K}, {2, 3, 4, 6, 7}, {2, 3, 4, 6, 8}, {2, 3, 4, 6, 9}, {2, 3, 4, 6, 10}, {2, 3, 4, 6, J}, {2, 3, 4, 6, Q}, {2, 3, 4, 6, K}, {2, 3, 4, 7, 8}, {2, 3, 4, 7, 9}, {2, 3, 4, 7, 10}, {2, 3, 4, 7, J}, {2, 3, 4, 7, Q}, {2, 3, 4, 7, K}, {2, 3, 4, 8, 9}, {2, 3, 4, 8, 10}, {2, 3, 4, 8, J}, 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9, 10, K}, {5, 7, 9, J, Q}, {5, 7, 9, J, K}, {5, 7, 9, Q, K}, {5, 7, 10, J, Q}, {5, 7, 10, J, K}, {5, 7, 10, Q, K}, {5, 7, J, Q, K}, {5, 8, 9, 10, J}, {5, 8, 9, 10, Q}, {5, 8, 9, 10, K}, {5, 8, 9, J, Q}, {5, 8, 9, J, K}, {5, 8, 9, Q, K}, {5, 8, 10, J, Q}, {5, 8, 10, J, K}, {5, 8, 10, Q, K}, {5, 8, J, Q, K}, {5, 9, 10, J, Q}, {5, 9, 10, J, K}, {5, 9, 10, Q, K}, {5, 9, J, Q, K}, {5, 10, J, Q, K}, {6, 7, 8, 9, 10}, {6, 7, 8, 9, J}, {6, 7, 8, 9, Q}, {6, 7, 8, 9, K}, {6, 7, 8, 10, J}, {6, 7, 8, 10, Q}, {6, 7, 8, 10, K}, {6, 7, 8, J, Q}, {6, 7, 8, J, K}, {6, 7, 8, Q, K}, {6, 7, 9, 10, J}, {6, 7, 9, 10, Q}, {6, 7, 9, 10, K}, {6, 7, 9, J, Q}, {6, 7, 9, J, K}, {6, 7, 9, Q, K}, {6, 7, 10, J, Q}, {6, 7, 10, J, K}, {6, 7, 10, Q, K}, {6, 7, J, Q, K}, {6, 8, 9, 10, J}, {6, 8, 9, 10, Q}, {6, 8, 9, 10, K}, {6, 8, 9, J, Q}, {6, 8, 9, J, K}, {6, 8, 9, Q, K}, {6, 8, 10, J, Q}, {6, 8, 10, J, K}, {6, 8, 10, Q, K}, {6, 8, J, Q, K}, {6, 9, 10, J, Q}, {6, 9, 10, J, K}, {6, 9, 10, Q, K}, {6, 9, J, Q, K}, {6, 10, J, Q, K}, {7, 8, 9, 10, J}, {7, 8, 9, 10, Q}, {7, 8, 9, 10, K}, {7, 8, 9, J, Q}, {7, 8, 9, J, K}, {7, 8, 9, Q, K}, {7, 8, 10, J, Q}, {7, 8, 10, J, K}, {7, 8, 10, Q, K}, {7, 8, J, Q, K}, {7, 9, 10, J, Q}, {7, 9, 10, J, K}, {7, 9, 10, Q, K}, {7, 9, J, Q, K}, {7, 10, J, Q, K}, {8, 9, 10, J, Q}, {8, 9, 10, J, K}, {8, 9, 10, Q, K}, {8, 9, J, Q, K}, {8, 10, J, Q, K}, {9, 10, J, Q, K}}

Length[Subsets[{A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K}, {5}]]

Out[180]= 5148

One pair:

32kort:

In[135]:= deck = Sort[Join[Range[7, 14], Range[7, 14], Range[7, 14], Range[7, 14]]]; hands = Subsets[deck, {5}]; onePairQ32[{, x, x, }] := True; onePairQ32[{, x, x, x, _}] := False; onePairQ32[{, x, x, , y, y, ___} /; x != y] := False; onePairQ32[{___}] := False; Count[hands, _?onePairQ32]

Out[141]= 107520

52kort:

In deck = Sort[Join[Range[1, 13], Range[1, 13], Range[1, 13], Range[1, 13]]]; hands = Subsets[deck, {5}]; onePairQ52[{, x, x, }] := True; onePairQ52[{, x, x, x, _}] := False; onePairQ52[{, x, x, , y, y, ___} /; x != y] := False; onePairQ52[{___}] := False; Count[hands, _?onePairQ52]

Out[134]= 1098240

Two pairs:

32kort:

In[8]:= deck = Sort[Join[Range[7, 14], Range[7, 14], Range[7, 14], Range[7, 14]]]; hands = Subsets[deck, {5}]; TwoPairQ32[{, x, x, }] := False; TwoPairQ32[{, x, x, x, _}] := False; TwoPairQ32[{, x, x, , y, y, ___} /; x != y] := True; TwoPairQ32[{___}] := False; Count[hands, _?TwoPairQ32]

Out[14]= 24192

52kort:

In[15]:= deck = Sort[Join[Range[1, 13], Range[1, 13], Range[1, 13], Range[1, 13]]]; hands = Subsets[deck, {5}]; TwoPairQ52[{, x, x, }] := False; TwoPairQ52[{, x, x, x, _}] := False; TwoPairQ52[{, x, x, , y, y, ___} /; x != y] := True; TwoPairQ52[{___}] := False; Count[hands, _?TwoPairQ52]

Out[21]= 123552

Three of a kind:

32kort:

In[22]:= deck = Sort[Join[Range[7, 14], Range[7, 14], Range[7, 14], Range[7, 14]]]; hands = Subsets[deck, {5}]; ThreeOfKindQ32[{, x, x, }] := False; ThreeOfKindQ32[{, x, x, x, _}] := True; ThreeOfKindQ32[{, x, x, , y, y, ___} /; x != y] := False; ThreeOfKindQ32[{___}] := False; Count[hands, _?ThreeOfKindQ32]

Out[28]= 12320

52kort:

In[142]:= deck = Sort[Join[Range[1, 13], Range[1, 13], Range[1, 13], Range[1, 13]]]; hands = Subsets[deck, {5}]; ThreeOfKindQ52[{, x, x, }] := False; ThreeOfKindQ52[{, x, x, x, _}] := True; ThreeOfKindQ52[{, x, x, , y, y, ___} /; x != y] := False; ThreeOfKindQ52[{___}] := False; Count[hands, _?ThreeOfKindQ52]

Out[148]= 59280

pairQ[{x, x, y, y} /; x != y] := False; (* two pairs *) pairQ[{, x, x, x, _}] := False; (* three of a kind *) pairQ[{, x, x, }] := True; (* a pair *) pairQ[{___}] := False (* else *)

Four of a kind:

32kort:

In[36]:= deck = Sort[Join[Range[7, 14], Range[7, 14], Range[7, 14], Range[7, 14]]]; hands = Subsets[deck, {5}]; FourOfKindQ32[{, x, x, }] := False; FourOfKindQ32[{, x, x, x, _}] := False; FourOfKindQ32[{, x, x, , y, y, ___} /; x != y] := False; FourOfKindQ32[{___}] := False; FourOfKindQ32[{, x, x, x, x, }] := True; Count[hands, _?FourOfKindQ32]

Out[43]= 224

52kort:

In[44]:= deck = Sort[Join[Range[1, 13], Range[1, 13], Range[1, 13], Range[1, 13]]]; hands = Subsets[deck, {5}]; FourOfKindQ52[{, x, x, }] := False; FourOfKindQ52[{, x, x, x, _}] := False; FourOfKindQ52[{, x, x, , y, y, ___} /; x != y] := False; FourOfKindQ52[{___}] := False; FourOfKindQ52[{, x, x, x, x, }] := True; Count[hands, _?FourOfKindQ52]

Out[51]= 624

Full house:

32kort:

In[98]:= deck = Sort[Join[Range[7, 14], Range[7, 14], Range[7, 14], Range[7, 14]]]; hands = Subsets[deck, {5}]; FullhouseQ32[{, x, x, }] := False; FullhouseQ32[{, x, x, x, _}] := False; FullhouseQ32[{, x, x, , y, y, ___} /; x != y] := False; FullhouseQ32[{___}] := False; FullhouseQ32[{x, x, x, y, y_} /; x != y] := True; Count[hands, _?FullhouseQ32]

Out[105]= 672

52kort:

In[157]:= deck = Sort[Join[Range[1, 13], Range[1, 13], Range[1, 13], Range[1, 13]]]; hands = Subsets[deck, {5}]; FullhouseQ52[{, x, x, }] := False; FullhouseQ52[{, x, x, x, _}] := False; FullhouseQ52[{, x, x, , y, y, ___} /; x != y] := False; FullhouseQ52[{___}] := False; FullhouseQ52[{x, x, y, y, y_} /; x != y] := True; Count[hands, _?FullhouseQ52]

Out[164]= 1872

Nothing:

32kort:

In[75]:= deck = Sort[Join[Range[7, 14], Range[7, 14], Range[7, 14], Range[7, 14]]]; hands = Subsets[deck, {5}]; NothingQ32[{, x, x, }] := False; NothingQ32[{, x, x, x, _}] := False; NothingQ32[{, x, x, , y, y, ___} /; x != y] := False; NothingQ32[{___}] := True; Count[hands, _?NothingQ32]

Out[81]= 57344

52kort:

In[165]:= deck = Sort[Join[Range[1, 13], Range[1, 13], Range[1, 13], Range[1, 13]]]; hands = Subsets[deck, {5}]; NothingQ52[{, x, x, }] := False; NothingQ52[{, x, x, x, _}] := False; NothingQ52[{, x, x, , y, y, ___} /; x != y] := False; NothingQ52[{___}] := True; Count[hands, _?NothingQ52]

Out[171]= 1317888

What is wrong with my solution ? please help me as soon as possible