Module 3 · Lesson 12 of 39 · Free

A deck in numbers: 52 cards, 1,326 starting hands

Course by Hernan Alejandro Gajardo Morales

The deck can deal you 1,326 different pairs of hole cards, but only 169 kinds of hand that play differently. A♠ K♦ and A♥ K♣ are two different hands of the same kind: before the flop the suits matter only when your two cards share one. Counting how many ways each kind can be dealt is the first step from cards to odds.

The hand

Your hole cards: A♠ K♦, an ace and a king of different suits. Before the flop they play exactly like A♥ K♣ or A♣ K♠.

Predict first

In how many ways can the deck deal you an ace and a king?

1. 8: there are 4 aces and 4 kings
2. 16: each of 4 aces with each of 4 kings
3. 12: the two suits must differ

Working it out

Count the deck first: 13 ranks × 4 suits = 52 cards. Your first card can be any of the 52 and your second any of the other 51, 52 × 51 = 2,652 ways, but that counts every hand twice: A♠ then K♦, and K♦ then A♠. So there are 52 × 51 ÷ 2 = 1,326 starting hands. A pair takes two of the four cards of one rank, 4 × 3 ÷ 2 = 6 combinations, and the 13 pairs make 13 × 6 = 78 hands. Two different ranks can be picked in 13 × 12 ÷ 2 = 78 ways, and each comes suited in 4 combinations, one per suit, and offsuit in 4 × 3 = 12. In total 78 + 78 × 4 + 78 × 12 = 78 + 312 + 936 = 1,326.

Count any group of hands the same way, kind by kind. Hands with at least one ace are quicker to count from the other side: hands with no ace use only the other 48 cards, 48 × 47 ÷ 2 = 1,128, so 1,326 − 1,128 = 198 hands hold at least one ace. On the 13 × 13 grid of starting hands, pairs run down the diagonal, suited hands sit above it and offsuit hands below it. Each cell is one kind, but the cells are not equal: an offsuit cell holds 12 hands, a pair 6 and a suited cell only 4.

Formula

C(52, 2) = 52 × 51 ÷ 2 = 1,326 starting hands, where C(n, k) counts the ways to choose k cards from n. One kind: pair 4 × 3 ÷ 2 = 6, suited 4, offsuit 4 × 3 = 12

Check your understanding

1. How many of the 1,326 starting hands hold at least one heart?

   Show the answer

   Answer: 585

   Hands without a heart use only the other 39 cards: 39 × 38 ÷ 2 = 741. So 1,326 − 741 = 585 hands hold at least one heart.

2. A player raises before the flop only with QQ, AKs and AQo. How many combinations is that?

   Show the answer

   Answer: 22

   QQ is a pair, 6. AKs is suited, 4. AQo is offsuit, 12. In total 6 + 4 + 12 = 22.

3. J♠ T♠ and J♥ T♥: what are they?

   1. Two combinations of one kind, JTs
   2. Two different kinds of hand
   3. Two names for the same combination
   Show the answer

   Answer: Two combinations of one kind, JTs

   Both are a jack and a ten of one suit: two of the 4 combinations of JTs, which is one of the 169 kinds.