Counting poker hands

Counts 5-card poker hands and a few hand types, and cross-checks the count against brute-force enumeration for a smaller deck.

from mathematicskit.combinatorics import combinations_count, generate_combinations, multinomial_coefficient

Total 5-card hands from a standard 52-card deck

total_hands = combinations_count(52, 5)
print(f"total 5-card hands: {total_hands}")

total 5-card hands: 2598960

Number of ways to deal a full house (3 of one rank, 2 of another)

full_houses = combinations_count(13, 1) * combinations_count(4, 3) * combinations_count(12, 1) * combinations_count(4, 2)
print(f"full houses: {full_houses} (probability {full_houses / total_hands:.6f})")

full houses: 3744 (probability 0.001441)

Cross-check via brute-force enumeration on a small 8-card deck

small_deck = list(range(8))
hands = generate_combinations(small_deck, r=3)
print(f"\nC(8,3) = {combinations_count(8, 3)}, generated {len(hands)} hands")

C(8,3) = 56, generated 56 hands

Multinomial coefficient: ways to deal a 13-card deck into 4 equal hands

deals = multinomial_coefficient(12, [3, 3, 3, 3])
print(f"ways to split 12 cards into 4 groups of 3: {deals}")

ways to split 12 cards into 4 groups of 3: 369600