Combinations in Gambling: How to Count Hands (nCr)

A combination counts the ways to choose r items from n items when the order of the choice does not matter. It is the tool behind nearly every card and lottery probability. A 52-card deck holds 1,326 possible two-card hands and 2,598,960 possible five-card hands, and both numbers come from one formula: nCr = n! ÷ (r! × (n − r)!).

This matters because a probability is favorable outcomes divided by total outcomes, as explained in probability in gambling, and the difficult part is nearly always the counting. Counting the ways to be dealt a pair of aces, a flush or three winning lottery numbers by hand is slow and easy to get wrong. Combinations turn each of those counts into a short, checkable calculation.

This guide explains when order matters and when it does not, shows the formula with every symbol defined, and works through two-card and five-card examples step by step. It then applies the same counting to a hypothetical lottery and lists the mistakes that most often produce a wrong count. Every result below is a long-run proportion of equally likely deals, not a prediction about any single hand.

What a Combination Counts (and What a Permutation Counts)

Suppose you are dealt two cards from a fresh deck. The hand “ace of spades and king of hearts” is the same hand whether the ace arrived first or second. A combination ignores order, so those two arrival sequences count as one hand. That matches how card hands, lottery tickets and most gambling outcomes are judged: what you hold matters, not the sequence you received it in.

A permutation is the other kind of count. A permutation respects order, so “ace then king” and “king then ace” are two different results. Permutations are the right tool when position matters, such as a trifecta in which first, second and third place must all be called in sequence.

The practical question is always the same: would swapping two of the items give a result the game treats differently? If not, use combinations. Combinations are never larger than permutations for the same n and r, because each combination stands for several orderings of the same items. With two cards, each hand stands for 2 orderings; with five cards, each hand stands for 5! = 120 orderings.

One more term is needed. A factorial, written n!, is the product of every whole number from n down to 1, so 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention 0! = 1, which keeps the formula working when you choose all or none of the items.

The Combination Formula, Symbol by Symbol

The formula for the number of combinations is:

where:

n = the total number of items to choose from (52 cards, 49 lottery numbers).
r = how many items you choose (2 cards, 5 cards, 6 numbers).
n! = n factorial, the product of all whole numbers from n down to 1.
nCr = the number of different groups of r items, order ignored (read “n choose r”).

The formula comes from permutations. The permutation count nPr = n! ÷ (n − r)! counts ordered selections, so 52P2 = 52 × 51 = 2,652. Each unordered pair appears r! times among those ordered selections, because r items can be arranged r! ways. Dividing by r! removes the duplicates:

In practice you rarely compute full factorials. Cancel the long tail of the numerator first: 52C5 = (52 × 51 × 50 × 49 × 48) ÷ (5 × 4 × 3 × 2 × 1) = 311,875,200 ÷ 120 = 2,598,960. The same shortcut makes 6C3 = (6 × 5 × 4) ÷ (3 × 2 × 1) = 20 a one-line calculation.

Two assumptions sit behind every count in this article. Every combination must be equally likely, which holds for a properly shuffled single deck or a fair lottery draw, and cards are drawn without replacement, so no card can appear twice in one hand.

Worked Example: Two-Card Hands From One Deck

Suppose you are dealt two cards from a well-shuffled 52-card deck. What is the chance of being dealt two aces, and what is the chance of any pair?

Step 1. Count all two-card hands: 52C2 = (52 × 51) ÷ 2 = 1,326.

Step 2. Count the ways to pick two of the four aces: 4C2 = (4 × 3) ÷ 2 = 6.

Step 3. Divide: 6 ÷ 1,326 ≈ 0.004525, which is about 0.45%, or roughly 1 in 221.

Step 4. For any pair, first choose the rank (13 ways), then choose two of that rank’s four cards (6 ways): 13 × 6 = 78 pair hands.

Step 5. Divide: 78 ÷ 1,326 ≈ 0.0588, which is about 5.88%, or 1 in 17.

On average, about 1 two-card deal in 17 is a pair, and about 1 in 221 is a specific pair such as two aces. These are long-run proportions: someone could see three pairs in a row or none in fifty deals, and neither result would contradict the math.

Applying Combinations Across Cards and Lotteries

The same three-step routine works everywhere: count all equally likely outcomes, count the favorable ones, divide. The favorable count is usually a product of smaller combinations. Four of a kind in five cards, for example, needs one of 13 ranks (13 ways) and one of the 48 remaining cards as the fifth card (48 ways): 13 × 48 = 624 hands.

A blackjack natural in a two-card hand needs one of 4 aces and one of 16 ten-value cards, so 4 × 16 = 64 hands out of 1,326, which is 4.83%. That agrees with the multiplication-rule answer 2 × (4/52) × (16/51) ≈ 4.83%, a useful confirmation that both methods count the same thing.

Lotteries use the same idea at larger scale. In a hypothetical game where you pick 6 numbers from 49, there are 49C6 = 13,983,816 possible tickets. Real lotteries publish their own matrices, so always confirm the numbers on the official lottery page before applying this method to a real game.

Common Mistakes When Counting Combinations

The most frequent error is using permutations when order does not matter. Counting 2,652 two-card hands instead of 1,326 doubles the total and cuts every probability in half. Ask the swap question first: if reordering the items gives the same result, divide by r!.

A second error is forgetting that cards are not replaced. The chance of drawing a second ace is 3/51, not 4/52, because one ace has already left the deck. Combinations handle this automatically, but hand-built fractions often do not.

A third is counting a hand twice when categories overlap. A full house contains a pair, so adding “pair” and “full house” counts together over-counts; hand rankings are usually defined so that each hand falls in exactly one category.

The last is treating a probability as a promise. A 1-in-221 chance does not mean two aces will arrive in 221 deals. Each deal is independent, which is the point made in the gambler’s fallacy guide.

Quick Reference and How to Check Your Answer

Use this checklist each time you count outcomes:

Check that the complete set adds up. In two-card hands, 78 pair hands plus 78 × 16 = 1,248 non-pair hands equals 1,326, the full total. If a set of categories should cover every outcome, its counts must sum to the total and its probabilities must sum to 100%.

A second check is magnitude: a favorable count can never exceed the total, and a combination such as 52C5 should be a large whole number, never a fraction.

Related Concepts and Next Steps

Combinations extend the basic rules in probability in gambling, which introduces favorable-over-total counting and the multiplication rule. Once you can count outcomes, the next step is to attach payouts to them: the expected value formula guide shows how a probability and a payout combine into an average result per bet.

It is also worth reading about variance in gambling, since a rare outcome with a large payout produces wide swings even when the average is stable. Counting tells you how likely an outcome is; it does not change the fact that standard casino bets carry a negative expected value for the player, and no counting method alters that.