Binomial Coefficient Calculator

Work out binomial coefficient instantly with clear inputs, formula shown and shareable results.

About Binomial Coefficient Calculator

The binomial coefficient C(n, r) = n!/(r!(n−r)!) counts the ways to choose r items from n when order does not matter. Permutations P(n, r) = n!/(n−r)! count the same selections when order does matter, so P = C × r!.

Formulas

Combinations

C(n, r) = n! / (r!(n − r)!)

Permutations

P(n, r) = n! / (n − r)!

Worked example

With the default values below, Binomial Coefficient Calculator returns C(n, r) of 210. Change any input above and every figure here is recalculated in your browser.

Inputs used

n (total items)accepts 0–1,000: 10
r (items chosen)accepts 0–1,000: 4

Results

C(n, r): 210
P(n, r) — order matters: 5,040
C(n, n − r): 210
Sum of the whole row 2ⁿ: 1,024

Step by step

1Values used
2Combinations
3Permutations
4C(n, r)= 210
5P(n, r) — order matters= 5,040
6C(n, n − r)= 210
7Sum of the whole row 2ⁿ= 1,024

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is C(10, 4)?

210, while P(10, 4) = 5040 because the four chosen items can be ordered in 4! = 24 ways.

Why does C(n, r) = C(n, n−r)?

Choosing which r to include is the same as choosing which n − r to leave out.