Binomial Coefficient Calculator
The coefficient of xᵏ in (1 + x)ⁿ — the entry at row n, position k of Pascal's triangle — with its neighbours and the row sum.
The binomial coefficient C(n, k) counts the k-subsets of an n-set, and it is exactly the coefficient of xᵏ when (1 + x)ⁿ is expanded.
How the binomial coefficient calculator works
The binomial coefficient C(n, k) counts the k-subsets of an n-set, and it is exactly the coefficient of xᵏ when (1 + x)ⁿ is expanded. Pascal's rule says each entry is the sum of the two above it: C(n, k) = C(n−1, k−1) + C(n−1, k).
Every row sums to 2ⁿ, because summing over all k counts every subset of every size.
Formula: C(n, k) = n! / (k! (n − k)!)
Worked examples
| Inputs | C(n, k) | Note |
|---|---|---|
| Row 10, position 4 | 210 | 210 |
| The middle of row 6 | 20 | 20 |
| Row 52, position 5 | 2,598,960 | 2,598,960 |
FAQFrequently asked questions
What is a binomial coefficient?
C(n, k) — the number of k-element subsets of an n-element set, and the coefficient of xᵏ in (1 + x)ⁿ.
What is Pascal's triangle?
The array of these coefficients, each entry the sum of the two above it. Row 4 reads 1, 4, 6, 4, 1.
Why does each row sum to 2ⁿ?
Because summing over every k counts every subset of the n-set, and there are 2ⁿ of those.
What is C(n, 0)?
Always 1 — there is exactly one empty subset.
Where is the largest entry?
In the middle: C(n, ⌊n/2⌋). The row is symmetric about it.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.