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CalcivoTools

Combination Calculator : Calculate nCr Binomial Coefficients (Order Does Not Matter)

The Calcivo Combination Calculator computes the number of subsets of size r that can be chosen from a set of size n, where the order of selection does not matter. It calculates standard combinations (nCr) and combinations with replacement.

Interactive Calculator

100% Client-Side Engine
Combinations (\binom{10}{4})
210
Binomial Coefficient Steps
1. n! = 10! = 3628800
2. r! = 4! = 24
3. (n - r)! = (10 - 4)! = 6! = 720
4. nCr = 3628800 ÷ (24 × 720) = 210
5. Symmetry note: \binom{10}{4} = \binom{10}{6} = 210

Step-by-Step Examples

Selecting a 4-Person Committee from 10 Candidates

Order does not matter: C(10, 4).

Inputs: Total Pool (n): 10, Chosen (r): 4
Result: 210 Combinations

10! / (4! × 6!) = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1) = 5040 / 24 = 210.

How to Use This Tool

  1. Choose standard combinations (nCr) or combinations with replacement.
  2. Enter the total pool size (n).
  3. Enter the subset sample size (r).
  4. Click Calculate to see the binomial coefficient and cancellation steps.

Frequently Asked Questions

By the symmetry of combinations, choosing r items to include in a group is mathematically equivalent to choosing (n - r) items to leave out.

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