What is the Combination Calculator?
A combination is a selection of items from a larger set where the order does not matter and no item is repeated. This calculator computes nCr — the number of distinct groups of size r you can form from a set of n different items. It is a core tool in combinatorics, probability, lottery analysis, and statistics.
How to use it
Enter the total number of items n and the number you want to choose r. The calculator instantly returns how many unique combinations are possible. Make sure r is no larger than n; choosing more items than exist is not defined and returns zero.
The formula explained
The number of combinations is given by:
$$C(n, r) = \binom{n}{r} = \frac{n!}{r!\,\left(n - r\right)!}$$
Here "!" is the factorial (the product of all positive integers up to that number). Dividing by both r! and (n−r)! removes the duplicate orderings that permutations would count, since order is irrelevant in a combination. To stay accurate for large numbers, this tool uses the multiplicative form rather than computing huge factorials directly.
Worked example
How many ways can you pick 3 students from a class of 5? Using the formula: $$\frac{5!}{3! \times 2!} = \frac{120}{6 \times 2} = \frac{120}{12} = 10.$$ So there are 10 possible 3-person teams.
FAQ
What is the difference between combinations and permutations? In permutations order matters (AB ≠ BA), while in combinations it does not (AB = BA). Permutations are always greater than or equal to combinations for the same n and r.
What is nC0 or nCn? Both equal 1 — there is exactly one way to choose nothing, and one way to choose everything.
Does this allow repeated items? No. This is combinations without repetition, meaning each item can be selected at most once.
