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Combination Calculator (nCr)Calculate combinations (unordered selections) with or without repetition.

Combination Calculator (nCr) illustration
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Combination Calculator (nCr)

Calculate combinations (unordered selections) with or without repetition.

How to Use
1

Enter n and r

Input the total items (n) and items to choose (r).

2

Toggle Repetition

Enable "Allow repetition" if items can be chosen more than once.

3

View Result

See C(n,r) with interpretation of what the number means.

What Is Combination Calculator (nCr)?

A combination is an unordered selection of items from a larger set. C(n,r), also written as "n choose r" or the binomial coefficient (n r), equals n!/(r!(n−r)!). Unlike permutations, combinations do not consider order — selecting ABC is the same as selecting BAC. This concept answers questions like "how many ways can I choose 3 books from a shelf of 10?" — C(10,3) = 120. The calculator also supports combinations with repetition (multichoose), using the formula C(n+r−1, r), for scenarios where items can be selected multiple times. Binomial coefficients appear in Pascal's triangle, the binomial theorem, probability distributions, and combinatorial identities throughout mathematics.

Why Use Our Combination Calculator (nCr)?

  • Calculates both standard and with-repetition combinations
  • Uses BigInt for exact results with large inputs
  • Shows clear interpretation of what the result means
  • Distinguishes from permutations to avoid common confusion
  • Supports n up to 1000

Common Use Cases

Lottery Odds

Calculate the odds of winning by finding C(n,r) for lottery number selections.

Committee Formation

Find how many ways to form a committee of r people from n candidates.

Menu Combinations

Calculate how many meal combinations are possible from a set of options.

Sampling

Determine the number of possible samples in statistical sampling without replacement.

Technical Guide

The combination formula C(n,r) = n!/(r!(n−r)!) counts unordered selections. It equals P(n,r)/r! since each combination corresponds to r! permutations. Key properties: C(n,0) = C(n,n) = 1, C(n,r) = C(n, n−r) (symmetry), and Pascal's rule: C(n,r) = C(n−1,r−1) + C(n−1,r). Combinations with repetition use the "stars and bars" formula: C(n+r−1, r), which counts the ways to place r identical balls into n distinct boxes. The binomial theorem states (a+b)^n = Σ C(n,k) × a^(n-k) × b^k for k from 0 to n, making binomial coefficients the expansion coefficients. This calculator uses BigInt division of factorials for exact computation.

Tips & Best Practices

  • 1
    C(n,r) = C(n, n-r) — choosing what to include is equivalent to choosing what to exclude
  • 2
    Combinations are always fewer than or equal to permutations for the same n,r
  • 3
    C(n,2) = n(n-1)/2 — a useful shortcut for choosing pairs
  • 4
    Use combinations with repetition when items can be selected multiple times
  • 5
    The sum of all C(n,k) for k=0 to n equals 2^n

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Frequently Asked Questions

QWhat is the difference between combinations and permutations?
Combinations ignore order (choosing ABC = choosing BAC), while permutations consider order (ABC ≠ BAC). Use combinations when order doesn't matter.
QWhat is "n choose r"?
It's another way to say C(n,r) — the number of ways to choose r items from n items without considering order. Also called a binomial coefficient.
QWhat are combinations with repetition?
When items can be chosen more than once. For example, choosing 3 scoops from 5 ice cream flavors allows repeats. The formula is C(n+r-1, r).
QWhy is C(n,0) = 1?
There is exactly one way to choose nothing from any set — don't pick anything. Similarly, C(n,n) = 1 because there's only one way to choose everything.
QHow are combinations related to Pascal's triangle?
Each entry in Pascal's triangle is C(n,r), where n is the row and r is the position. Each entry equals the sum of the two entries above it.

About Combination Calculator (nCr)

Combination Calculator (nCr) is a free online tool from FreeToolkit.ai. All processing happens directly in your browser — your data never leaves your device. No registration required. No ads. Just fast, reliable tools.