Combination Calculator

About the Combination Calculator

A combination is a selection of items where the order does not matter. Picking three friends to share a taxi, choosing six lottery numbers or being dealt a poker hand are all combinations: the same group chosen in a different order counts only once. Counting combinations is central to probability, because the chance of a particular outcome is often one divided by the number of possible combinations.

This combination calculator finds nCr, the number of ways to choose r items from n, either without repetition (each item used at most once) or with repetition (items may be chosen more than once). It works with exact whole numbers of any practical size, so lottery- and card-sized answers keep every digit. It also shows the chance of one particular selection and how many permutations there would be if order mattered.

How to Use the Combination Calculator

Enter the total number of items, n.

Enter the number to choose, r.

Choose whether repetition is allowed.

The number of combinations appears with the formula and the related counts.

The Formulas

  without repetition:  C(n, r) = n! ÷ ( r! × (n − r)! )
  with repetition:     C(n + r − 1, r)

  where n! = n × (n − 1) × … × 2 × 1, and 0! = 1

The notations nCr, C(n, r) and "n choose r" all mean the same thing.

Step-by-Step Example

How many ways can 3 people be chosen from 10?

  C(10, 3) = 10! ÷ (3! × 7!)
           = (10 × 9 × 8) ÷ (3 × 2 × 1)
           = 720 ÷ 6
           = 120

There are 120 possible groups of three. The 7! cancels out, which is why only the top three terms of 10! are needed.

Step-by-Step Example: With Repetition

How many ways can you choose 3 scoops from 5 ice cream flavours, if flavours can repeat?

  C(5 + 3 − 1, 3) = C(7, 3) = (7 × 6 × 5) ÷ (3 × 2 × 1) = 35

There are 35 different bowls, from three scoops of one flavour to three different flavours. This is often called the "stars and bars" method.

Famous Combination Counts

  lottery, 6 from 49               13,983,816
  poker hands, 5 from 52            2,598,960
  bridge hands, 13 from 52    635,013,559,600
  committee of 3 from 10                  120

A single 6-from-49 lottery ticket therefore has about a 1 in 14 million chance of the jackpot. Every possible hand in poker is one of 2,598,960, which is how the odds of flushes, full houses and royal flushes are worked out.

Combinations Versus Permutations

The key question is whether order matters. Choosing 3 of 10 people for a committee gives 120 combinations. Choosing a chair, secretary and treasurer from the same 10 gives 720 permutations, because each group of three can be arranged in 3! = 6 ways, and 120 × 6 = 720. In general, the number of permutations equals the number of combinations times r!. If swapping two chosen items gives a different result, you need permutations; if it gives the same result, you need combinations.

Why the Numbers Grow So Fast

Combination counts grow astonishingly quickly. Choosing 2 from 10 gives 45, choosing 5 from 20 gives 15,504, and choosing 10 from 40 gives 847,660,528. The largest count in each row is always near the middle, when r is about half of n. This explosive growth is why exhaustive searches become impossible so quickly: trying every possible team of 11 players from a squad of 25 would mean checking 4,457,400 line-ups, and every doubling of the squad multiplies that figure many times over. It is also why lotteries can offer huge jackpots — even a modest-looking game has millions of possible tickets.

Symmetry and Pascal's Triangle

Choosing 3 items to keep is the same as choosing 7 to leave out, so C(10, 3) = C(10, 7) = 120. This symmetry shows up in Pascal's triangle, where each row lists C(n, 0), C(n, 1) and so on, and each number is the sum of the two above it. The same numbers are the coefficients in the expansion of (a + b)^n, which is why they are also called binomial coefficients and appear in the binomial distribution.

Combinations in Probability

Combinations turn counting into probability. The chance of being dealt four aces in a five-card hand is the number of such hands divided by all hands: the four aces plus any one of the other 48 cards gives 48 hands, so the probability is 48 ÷ 2,598,960, about 1 in 54,145. The same approach finds the chance of matching some but not all lottery numbers, or of a quality inspector's sample containing a certain number of faulty parts.

Understanding Your Result

The headline is the number of combinations.

The formula line shows the calculation.

The chance of one selection line gives the probability of one particular group if all are equally likely.

The if order mattered line gives the matching number of permutations.

The worth knowing line adds a related fact about the count.

When Should You Use This Calculator?

Use it to count possible groups, teams, hands or lottery tickets.

Use it to work out probabilities in card games and lotteries.

Use it for homework on combinations and binomial coefficients.

Use it to count menu choices, sample selections or test designs.

Common Mistakes

Using combinations when order matters. Rankings and passwords need permutations.

Forgetting that 0! = 1. C(n, 0) and C(n, n) both equal 1.

Allowing repetition by mistake. Most selections of people or cards cannot repeat.

Multiplying out huge factorials. Cancel first, or let the calculator work exactly.

Assuming all combinations are equally likely. The probability rule needs a fair, random selection.

Mixing up n and r. The total number of items is n and the number chosen is r; C(3, 10) is not a sensible question without repetition.