Calculate nCr combinations instantly with step-by-step working. Find how many ways r items can be selected from n items when order does not matter — free, fast, and mobile-friendly.
Find how many ways r items can be chosen from n items, order ignored.
Choosing 3 people out of 10 for a committee, dealing 5 cards from a 52-card deck, or working out your odds in a lottery draw — all of these are combination problems. A combination counts the number of ways to select r items from a group of n items when the order of selection does not matter. Our free Combinations Calculator (nCr) gives you the answer instantly, complete with step-by-step working so you can see exactly how the result is derived. Whether you are a student learning combinatorics, a teacher preparing examples, a poker player computing hand probabilities, or anyone who needs to count selections quickly, this tool does the heavy lifting for you.
Unlike manual calculation, which becomes painful the moment factorials grow (10! is already 3,628,800), the calculator uses an efficient multiplicative method that keeps intermediate values small and returns exact results for everyday values of n and r. It also validates your inputs — n and r must be whole numbers, r cannot exceed n — and explains the substitution into the formula line by line. Everything runs in your browser; nothing you type is sent anywhere.
Using the calculator takes less than ten seconds. Follow these steps:
A quick sanity check before you trust any combination result: ask yourself whether rearranging the chosen items would create a genuinely different outcome. If the answer is no — a committee of Alice, Ben, and Cara is the same committee as Cara, Alice, and Ben — you are in combination territory and nCr is the right formula. If the answer is yes — first, second, and third place in a race are different outcomes — you need permutations (nPr) instead.
In combinatorics, a combination is a selection of items from a larger set where the order of selection is irrelevant and each item can be chosen at most once (selection is without replacement). The notation C(n, r), also written as nCr or as the binomial coefficient “n choose r”, answers one question: out of n distinct items, how many different groups of r items can be formed?
The idea is easiest to grasp with a tiny example. Suppose you have 4 fruits — an apple, a banana, a cherry, and a date — and you want to pick 2. List them out: apple-banana, apple-cherry, apple-date, banana-cherry, banana-date, cherry-date. That is 6 pairs. Notice that apple-banana and banana-apple are not counted separately, because the order does not change the pair. So C(4, 2) = 6. If order did matter, there would be 12 arrangements — that is the permutation count, P(4, 2) = 12, exactly double, because every unordered pair can be arranged in 2! = 2 orders.
This relationship is the key to the whole formula. Start with permutations: there are n × (n − 1) × … × (n − r + 1) ways to pick r items in order from n items. But each unordered group of r items gets counted once for every possible ordering, and r items can be ordered in r! (r factorial) ways. To collapse all those duplicate orderings into single combinations, divide by r!. The result is the combinations formula, one of the most useful counting tools in mathematics, statistics, and probability.
Combinations show up far beyond the classroom. Lottery odds are pure combinations: a 6-from-49 lottery has C(49, 6) = 13,983,816 possible tickets, which is why the jackpot is so hard to win. Card games run on combinations: there are C(52, 5) = 2,598,960 possible five-card poker hands, and every hand probability you have ever seen is a ratio of two such counts. Quality control uses them when inspectors sample r items from a production batch of n. Team selection, menu planning, survey sampling, and genetics (choosing which genes combine) all lean on the same counting idea. Anywhere you choose a subset and the arrangement inside the subset does not matter, nCr is quietly doing the work.
C(n, r) = n! / (r! × (n − r)!)
n = total items • r = items chosen • ! = factorial • requires 0 ≤ r ≤ n
Here n! (n factorial) means n × (n − 1) × (n − 2) × … × 2 × 1, with the special definition 0! = 1. The formula has a satisfying symmetry: choosing r items to take is the same as choosing n − r items to leave behind, so C(n, r) always equals C(n, n − r). That is why C(10, 8) = C(10, 2) = 45, and why the calculator internally works with the smaller of r and n − r to keep the arithmetic fast.
In practice you rarely expand the full factorials, because most terms cancel. For C(10, 3), writing out 10! / (3! × 7!) lets the entire 7! cancel against the tail of 10!, leaving only (10 × 9 × 8) / (3 × 2 × 1) = 720 / 6 = 120. The calculator shows exactly this cancelled-down form in its step-by-step panel, which is also the fastest way to compute combinations by hand and the method the widget uses internally to avoid overflow on large values of n.
Example 1: Committee of 3 from 10 people — C(10, 3)
A club with 10 members needs a 3-person committee, and nobody cares who is listed first. Substitute into the formula: C(10, 3) = 10! / (3! × 7!). Cancel 7! from top and bottom to get (10 × 9 × 8) / (3 × 2 × 1). The numerator is 720 and the denominator is 6, so the answer is 120 possible committees. Try it in the calculator: n = 10, r = 3, and the steps panel reproduces this exact working.
Example 2: Five-card poker hands — C(52, 5)
How many distinct 5-card hands can be dealt from a standard 52-card deck? Order does not matter — the same five cards in a different dealing sequence is the same hand. C(52, 5) = 52! / (5! × 47!) = (52 × 51 × 50 × 49 × 48) / (5 × 4 × 3 × 2 × 1) = 311,875,200 / 120 = 2,598,960 hands. This number is the denominator behind every poker probability: a royal flush, for instance, can occur in only 4 of those hands, giving odds of 4 / 2,598,960, roughly 1 in 649,740.
Example 3: Edge cases — C(7, 0), C(7, 7), and C(7, 1)
Choosing nothing from 7 items: C(7, 0) = 7! / (0! × 7!) = 1, because 0! = 1. There is exactly one way to choose nothing — the empty selection. Choosing everything: C(7, 7) = 7! / (7! × 0!) = 1, again one way. Choosing a single item: C(7, 1) = 7! / (1! × 6!) = 7 / 1 = 7, one per item. These edge cases confirm the formula behaves sensibly at the boundaries, and the calculator handles all of them — try n = 7, r = 0 to see the “exactly one way” explanation in the steps panel.
Lottery odds, honestly computed
A typical 6-from-49 lottery asks you to pick 6 numbers out of 49, and the draw order is irrelevant. The total number of possible tickets is C(49, 6) = 49! / (6! × 43!) = 13,983,816. Buying one ticket therefore gives you a 1-in-13,983,816 shot at the jackpot. The calculator confirms this instantly (n = 49, r = 6), and the number explains why lottery jackpots roll over so often — the combination count is simply enormous relative to the tickets sold.
Building a project team
A manager must pick 4 engineers out of 12 for a project team. Since roles are not yet assigned, this is a combination: C(12, 4) = 12! / (4! × 8!) = (12 × 11 × 10 × 9) / (4 × 3 × 2 × 1) = 11,880 / 24 = 495 possible teams. If two specific engineers refuse to work together, you can refine the count by subtracting the teams containing both — C(10, 2) = 45 — leaving 450 valid teams. Combination arithmetic like this turns vague “lots of options” thinking into precise numbers for planning.
Pizza toppings and menu design
A pizzeria offers 15 toppings and lets customers choose any 3. The number of distinct 3-topping pizzas is C(15, 3) = (15 × 14 × 13) / (3 × 2 × 1) = 2,730 / 6 = 455 combinations. Restaurant owners use exactly this math to estimate menu variety, and it is a great classroom example because students can verify small cases by listing. Note the contrast: if the pizzeria layered toppings in a specific order that mattered, you would need permutations instead — 2,730 arrangements, six times as many.
Quality-control sampling
An inspector pulls 5 units at random from a batch of 200 to test for defects. The number of possible samples is C(200, 5) = 2,535,650,040 — over 2.5 billion. Statisticians use this count as the denominator when computing the probability that a sample catches at least one defective unit (the hypergeometric distribution). Without a calculator, C(200, 5) is essentially uncomputable by hand; with one, it takes a second.
1. Using combinations when order matters. This is the single most frequent error. Assigning president, vice-president, and treasurer from 10 people is not C(10, 3) = 120 — the roles make order matter, so it is P(10, 3) = 720. Always ask: would reshuffling the chosen items create a different outcome? If yes, use permutations.
2. Forgetting that 0! = 1. Students often stall on C(n, 0) or C(n, n) because “zero factorial” looks undefined. By definition 0! = 1, which makes both equal 1 — exactly one way to choose nothing, exactly one way to choose everything. The definition is not arbitrary; it keeps the formula consistent at the boundaries.
3. Trying r larger than n. C(5, 8) is meaningless — you cannot choose 8 distinct items from 5. The calculator rejects this with a clear message, but on paper it is a sign you have mixed up which number is n and which is r. Remember: n is the big pool, r is the handful you take from it.
4. Expanding full factorials instead of cancelling first. Computing 20! (= 2,432,902,008,176,640,000) by hand to find C(20, 3) invites arithmetic errors. Cancel first: (20 × 19 × 18) / (3 × 2 × 1) = 6,840 / 6 = 1,140. The calculator’s steps panel demonstrates this cancelled form, which is both faster and far less error-prone.
5. Confusing “with replacement” scenarios. The nCr formula assumes each item is chosen at most once. If repeats are allowed — like rolling a die three times, where you can roll a 4 twice — the count is different: C(n + r − 1, r), the “stars and bars” formula. Make sure your problem really is selection without replacement before applying nCr.
Keep exploring counting and probability with these free tools: the Percentage Calculator for turning combination counts into probabilities and odds; the Probability Calculator for working with favorable versus total outcomes; the Fraction Calculator for simplifying ratios like 4 / 2,598,960; the Ratio Calculator for comparing counts; and the Absolute Difference Calculator for measuring gaps between two values.
This page provides a free online Combinations Calculator (nCr) at Sigma Calculator. A combination C(n, r) = n! / (r! × (n − r)!) counts the distinct ways to choose r items from n distinct items without replacement when order does not matter; e.g., C(10, 3) = 120, C(52, 5) = 2,598,960, C(49, 6) = 13,983,816. Key facts: C(n, r) = C(n, n − r); C(n, 0) = C(n, n) = 1; 0! = 1. Canonical URL: https://sigmacalculator.net/math-calculators/combinations-calculator/. Last updated October 9, 2026.
nCr, written C(n, r) or “n choose r”, is the number of combinations of r items that can be selected from n distinct items when the order of selection does not matter and no item is repeated. It is computed as n! / (r! × (n − r)!). For example, C(10, 3) = 120, meaning there are 120 different 3-person groups you can form from 10 people.
nCr (combinations) counts selections where order does not matter, while nPr (permutations) counts arrangements where order does matter. They are related by nPr = nCr × r!, because every unordered group of r items can be arranged in r! orders. Example: choosing any 3 of 10 people is C(10, 3) = 120, but assigning them distinct roles (president, vice-president, treasurer) is P(10, 3) = 720.
Write C(n, r) = n! / (r! × (n − r)!), then cancel the larger factorial before multiplying. For C(12, 4): cancel 8! to get (12 × 11 × 10 × 9) / (4 × 3 × 2 × 1) = 11,880 / 24 = 495. Cancelling first keeps the numbers small — you never need to compute the full 12! = 479,001,600. The calculator above shows this cancelled-down working automatically.
Both equal 1. C(n, 0) = n! / (0! × n!) = 1 because there is exactly one way to choose nothing — the empty selection — and 0! is defined as 1. C(n, n) = n! / (n! × 0!) = 1 because there is exactly one way to take every item. These edge cases keep the formula consistent, and they match the symmetry rule C(n, r) = C(n, n − r).
No. C(n, r) is only defined for 0 ≤ r ≤ n, because you cannot choose more distinct items than exist. Asking for C(5, 8) — 8 items chosen from 5 — is meaningless, and the calculator will reject it with an error message. If you find yourself wanting r larger than n, double-check which value is the pool (n) and which is the selection (r); they are often swapped by mistake.
Combinations power lottery odds (a 6-from-49 draw has C(49, 6) = 13,983,816 possible tickets), card-game probabilities (2,598,960 five-card poker hands), team and committee selection, quality-control sampling, menu and product variety planning, and genetics. Anywhere you pick a subset and the internal arrangement does not matter, nCr gives the exact count.
Instead of computing giant factorials directly, the calculator uses the cancelled multiplicative form — multiplying n × (n − 1) × … while dividing by 1 × 2 × … × r step by step — so intermediate values stay small. It accepts n up to 1000 and returns exact integer results within JavaScript’s safe-integer range, switching to scientific notation for larger values, and warns you if a result is too large to display.
Every entry in Pascal’s triangle is a combination: the entry in row n, position r equals C(n, r). Each number is the sum of the two above it, which mirrors the identity C(n, r) = C(n − 1, r − 1) + C(n − 1, r). So the triangle is a visual table of “n choose r” values — row 4 reads 1, 4, 6, 4, 1, which are C(4,0) through C(4,4).
Cite this page as: “Combinations Calculator (nCr).” Sigma Calculator. https://sigmacalculator.net/math-calculators/combinations-calculator/
Last updated: October 9, 2026