Combination Calculator
Calculate the number of unordered selections of r items from n distinct items, where the order does not matter. Ideal for lottery draws, committees, and poker hands.
Last updated: May 2026
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About this calculator
A combination counts the number of ways to choose r items from a set of n distinct items when the order of selection does not matter. The formula is C(n, r) = n! / (r! × (n − r)!), often read as 'n choose r'. It is derived from the permutation count by dividing out the r! different orderings of each selected group, since combinations treat any rearrangement of the same items as identical: P(n, r) = C(n, r) × r!. This is why there are always fewer combinations than permutations for the same n and r. Combinations answer questions where only the membership of the group matters, not its arrangement — choosing a committee of 3 from 10 people, dealing a 5-card poker hand from 52, or picking 6 lottery numbers from 49. The values, known as binomial coefficients, also appear as the entries of Pascal's triangle and as the coefficients in the binomial expansion of (a + b)^n. Edge cases follow useful symmetry and identity rules: C(n, 0) = C(n, n) = 1 (one way to choose nothing or everything), C(n, 1) = n (n ways to choose a single item), and C(n, r) = C(n, n − r) (choosing r to include is the same as choosing n − r to exclude). The result is undefined or zero when r exceeds n. Like permutations, combination counts can become very large, though they grow more slowly than permutations because of the r! in the denominator.
How to use
Example 1 — choosing 3 from 10, order irrelevant. Enter n = 10, r = 3. C(10, 3) = 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120. Verify: this is the permutation count of 720 divided by the 6 orderings of each group of 3, giving 120 distinct committees. Example 2 — a 5-card poker hand from a 52-card deck. Enter n = 52, r = 5. C(52, 5) = 52! / (5! × 47!) = 2,598,960. Verify: this is the famous total number of distinct five-card poker hands, confirming the formula on a well-known result.
Frequently asked questions
When do I use a combination instead of a permutation?
Use a combination when the order of the selected items does not matter — only which items are chosen. A committee of three people is the same regardless of the order you pick them, so it is a combination; but awarding three ranked prizes is a permutation because order matters. Lottery draws, poker hands, and choosing a subset of options are all combinations. The quick test is to ask whether rearranging the chosen items creates a different outcome: if not, use a combination. Because combinations ignore order, their count is always smaller than the corresponding permutation count by a factor of r!.
What does 'n choose r' mean?
'n choose r' is the spoken name for the combination C(n, r), the number of ways to choose r items from n when order does not matter. It is written as a binomial coefficient and appears throughout mathematics. The phrase emphasizes the selection aspect: from a set of n, how many distinct groups of size r can you form? For example, '52 choose 5' is the number of possible five-card hands. The same numbers form the rows of Pascal's triangle and the coefficients when expanding (a + b)^n, linking combinations to algebra and probability. It is one of the most useful counting quantities in all of mathematics.
Why does C(n, r) equal C(n, n − r)?
This symmetry reflects a simple idea: choosing which r items to include is exactly equivalent to choosing which n − r items to leave out. Every selection of r items automatically determines the n − r items not selected, so the two counts must be equal. For example, choosing 3 of 10 people for a committee (120 ways) is the same as choosing 7 of 10 to exclude (also 120 ways). This identity is handy for computation: to find C(50, 48), you can instead compute the much easier C(50, 2). Recognizing the symmetry can save substantial arithmetic when r is close to n.
How are combinations used in probability?
Combinations are central to probability whenever you count equally likely outcomes that are unordered. To find the chance of an event, you often divide the number of favorable combinations by the total number of possible combinations. For instance, lottery odds are 1 divided by C(n, r), and the probability of a particular poker hand is the number of ways to make it divided by C(52, 5). Combinations also define the binomial distribution, which models the number of successes in repeated trials. Mastering combinations is therefore a prerequisite for solving a huge range of probability problems. They convert 'how many ways' into the building blocks of likelihood.
When should I NOT use a combination calculator?
Do not use it when order matters — for ranked selections, sequences, or arrangements, you need permutations instead, which give a larger count. It assumes the n items are distinct and that each is chosen at most once; if repetition is allowed or some items are identical, a different formula (combinations with repetition) applies. It is undefined when r is greater than n, since you cannot choose more items than exist. For very large n, be aware that exact results can exceed standard numeric precision, though the symmetry identity often helps. Always confirm that the selection is genuinely unordered before using combinations.