Week 2: Counting the deck

Combinatorics before probability — how many starting hands there really are, and why the answer depends on what you're counting

1 March 2027

Why counting comes first#

Every equity figure, every outs count, every pot-odds threshold later in this course reduces to a ratio of two counts: the number of ways the thing you want can happen, over the number of ways anything can happen. Get the counting wrong and no amount of probability theory rescues the answer — it was wrong before the division sign showed up. So before this course touches a single percentage, it spends a week on counting the deck properly.

Two ways to count a starting hand#

Ask “how many different starting hands are there in hold ‘em?” at a table and you’ll get two different correct-sounding answers, because the question is ambiguous between two things worth telling apart.

A combo is a specific pair of cards: the 3♣ and the 3♦ is a different combo from the 3♣ and the 3♥, even though a player would call both hands “a pair of threes.” The number of combos is exactly the number of ways to choose 2 cards from a 52-card deck, order not mattering:

C(52, 2) = 52 × 51 / 2 = 1,326 combos

A hand class collapses combos that play identically before the flop: all six ways to be dealt pocket threes are “the same hand” for strategy purposes, and so are all four ways to hold a suited ace-king. Counting classes means counting pairs, suited hands, and offsuit hands separately:

• Pairs. 13 ranks, and for each rank C(4,2) = 6 ways to choose the two suits, giving 13 × 6 = 78 pair combos, but each rank is only one class, so 13 pair classes.
• Suited hands. C(13,2) = 78 ways to choose two distinct ranks, and for each rank pair, 4 ways to choose a shared suit, giving 78 × 4 = 312 suited combos, one class per rank pair, so 78 suited classes.
• Offsuit hands. The same 78 rank pairs, but now 4 × 3 = 12 ways to give the two ranks different suits, giving 78 × 12 = 936 offsuit combos, again 78 classes.

Add the combo counts: 78 + 312 + 936 = 1,326, matching C(52, 2) exactly — a useful check that no case was double-counted or missed. Add the class counts: 13 + 78 + 78 = 169 hand classes.

A folk claim to formalize: “there are 169 starting hands”#

This is the sentence that motivated the section above, and it is a good example of a claim that is really two different claims wearing one number. Said about strategy — how many meaningfully different decisions a player can face preflop — 169 is the right count, because equity and correct play depend only on rank and suitedness pattern, not which physical suits are involved. Said about probability — how likely you are to be dealt a specific hand — 169 is the wrong count to divide by, because the 169 classes are not equally likely: a pair class has 6 combos behind it and a suited class has only 4, so P(pair) ≠ P(suited hand) even though “pair” and “suited hand” are each one class out of 169. The combo count, not the class count, is what a probability calculation needs.

Before next week#

Week 3 reuses this same counting machinery on the flop and turn rather than the two hole cards — that’s how the “rule of 4 and 2” from week 1 gets its exact version. Before then, be comfortable computing C(n, k) by hand for small n, since next week’s derivation leans on it directly.