Combinations: counting without listing
C(n, k) and why order never matters for cards.
6 minute read
How many two-card starting hands are there? You could deal them out one by one, or count: 52 choices for the first card, 51 for the second, and divide by 2 because A♠K♠ and K♠A♠ are the same hand. 52 × 51 ÷ 2 = 1,326.
Combinations
C(n, k) = n! ÷ (k! × (n − k)!)
The number of ways to choose k items from n when order does not matter.
- C(52, 2) = 1,326 starting hands
- C(4, 2) = 6 ways to hold a specific pocket pair
- C(52, 5) = 2,598,960 five-card hands
- C(47, 2) = 1,081 turn-and-river combinations from the flop
- C(48, 5) = 1,712,304 boards when two hands are known
Order matters for permutations (the sequence of the flop cards, say) but never for what a hand is. That is why every count on this site is a combination, and why "ways to miss" arguments work: the number of two-card run-outs that avoid your nine outs is C(38, 2) = 703, out of C(47, 2) = 1,081.
Click any hand. Suited above the diagonal, offsuit below, pairs on it.
Shading: brighter green is higher equity. Monte Carlo simulation, seeded xoshiro128**, 500,000 trials per hand, seed 20260901.
Suited
Ace-King suited
Equity against random hands
Multi-way, a hand’s share of the pot falls even when it is the favourite: a fair share against 8 opponents is 11.1%.
Head-to-head, exact
Every one of the 1,712,304 possible boards, enumerated for each matchup.
- vs Pocket aces…
- vs Pocket queens…
- vs Ace-king offsuit…
- vs Pocket sevens…
Quick check
How many combinations of pocket kings are there? Of king-queen suited? Of king-queen offsuit?
Show answer
6, 4 and 12.