What is a binomial coefficient?

The binomial coefficient C(n, k), also written ⁿCₖ or (n choose k), is the number that multiplies the aⁿ⁻ᵏbᵏ term when (a + b)ⁿ is expanded. It's calculated as n! / (k!(n - k)!).

How is this different from the Combination Calculator?

They use the same formula and give the same number, but this tool frames the result as a term in a binomial expansion (a + b)ⁿ, while the Combination Calculator frames it as choosing k items from a set of n — the same math, two different real-world uses.

What is the Binomial Theorem?

The Binomial Theorem states that (a + b)ⁿ = Σ C(n, k) aⁿ⁻ᵏ bᵏ, summed over k = 0 to n. Each term's coefficient is exactly the binomial coefficient C(n, k).

Why does the calculator show a Pascal's Triangle row?

Row n of Pascal's Triangle contains every coefficient C(n, 0) through C(n, n) in order. Viewing the whole row shows the symmetric pattern — C(n, k) always equals C(n, n - k) — and where your specific coefficient sits relative to its neighbors.

What happens if k is greater than n?

C(n, k) is undefined when k > n, since a binomial expansion of degree n only has terms for k from 0 to n. The calculator will show an error in this case.

What are C(n, 0) and C(n, n)?

Both always equal 1 — the first and last terms of any binomial expansion (aⁿ and bⁿ) always have a coefficient of 1.

You can enter values of n up to 1,000 and get an exact coefficient. The Pascal's Triangle row visualization is shown for n up to 20, since larger rows become impractical to display.

Where are binomial coefficients used besides algebra?

Binomial coefficients are the foundation of the binomial probability distribution, appear in combinatorics problems, and show up throughout computer science in algorithm analysis and counting problems.

Binomial Coefficient Calculator — Find C(n, k) and the Binomial Expansion Term

Our Binomial Coefficient Calculator finds C(n, k) — the coefficient that appears in front of the aⁿ⁻ᵏbᵏ term when you expand (a + b)ⁿ using the Binomial Theorem. Enter n and k to get the exact coefficient, the full expansion term, and the entire row of Pascal's Triangle for that n with your coefficient highlighted, so you can see exactly how it fits among its neighbors.

Quick Answer

The binomial coefficient C(n, k) is the number in front of the aⁿ⁻ᵏbᵏ term when (a + b)ⁿ is expanded, calculated as n! / (k!(n − k)!). Enter n and k below to get the exact coefficient, the full expansion term, and a Pascal's Triangle row view.

Exponent (n)

Term position (k)

Enter n and k, then click Calculate.

How to Use the Binomial Coefficient Calculator — Calculate C(n,k) Online

1
Enter the exponent n from the binomial expansion (a + b)ⁿ.
2
Enter the term position k (0 to n) you want the coefficient for.
3
Click 'Calculate' to find C(n, k) and the matching expansion term aⁿ⁻ᵏbᵏ.
4
For n up to 20, view the full Pascal's Triangle row with your coefficient highlighted.

Why Use Binomial Coefficient Calculator — Calculate C(n,k) Online?

The Binomial Theorem is one of the most-used identities in algebra and probability, but expanding (a + b)ⁿ by hand for anything past n = 4 or 5 quickly becomes tedious and error-prone. Our calculator gives you any single coefficient instantly — useful for checking one term of an expansion without writing out the whole thing — while the Pascal's Triangle row view shows the coefficient in context, making the symmetry and pattern of binomial coefficients visible at a glance.

Frequently Asked Questions

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