Binomial Theorem Expansion Calculator

The Binomial Theorem Expansion Calculator expands (a + b)ⁿ for any whole-number exponent up to 50, listing every term of the expansion along with its binomial coefficient, its calculated numeric coefficient, and its algebraic form. Enter term a, term b, and the exponent n to get the full expansion, a term-by-term breakdown table, and two running sums.

It's built for algebra and combinatorics students verifying binomial expansions by hand, teachers preparing worked examples, and anyone who needs an exact expansion at a exponent too large to expand by repeated multiplication.

How to use it

1. Enter term a — an integer coefficient plus an optional single-letter variable, e.g. 2x, -y, or 3.
2. Enter term b in the same format.
3. Set the exponent (n) with the slider, from 0 to 50.
4. Read the Expression and Full Expansion, scan the Terms table for each term's binomial coefficient and value, and check the two coefficient sums.

Results update automatically as you type. For example, a = 2x, b = 3, n = 4 expands to 16x^4 + 96x^3 + 216x^2 + 216x + 81.

The binomial theorem

For a whole number n, (a + b)ⁿ expands as:

(a + b)^n = Σ (k = 0 to n) C(n,k) · a^(n−k) · b^k

Each C(n,k) — read "n choose k" — is a binomial coefficient from row n of Pascal's triangle: C(n,k) = n! / (k! · (n−k)!). The calculator builds Pascal's triangle up to row n, then multiplies each coefficient by a^(n−k) · b^k to get every term's final numeric coefficient.

Why exact arithmetic matters at n = 50

A coefficient like 9 raised to the 50th power has 48 digits — far beyond what ordinary floating-point numbers can represent exactly (they lose precision past about 15–17 significant digits). Every step here — Pascal's triangle, each term's coefficient, and both running sums — is computed with arbitrary-precision integer arithmetic, so the results stay exact no matter how large the coefficients or exponent get.

The two coefficient sums

• Sum of Binomial Coefficients is C(n,0) + C(n,1) + ... + C(n,n), which always equals 2ⁿ — independent of what a and b actually are.
• Sum of All Coefficients is the sum of every term's calculated numeric coefficient (the binomial coefficient times the powers of a and b's own coefficients). Setting the variables to 1 shows why: it always equals (a + b)ⁿ evaluated with just the numeric coefficients — for 2x + 3 that's (2 + 3)^4 = 625, matching 16 + 96 + 216 + 216 + 81.

Combining like variables

If both terms use the same variable letter (e.g. x and x), the calculator merges their exponents into one power rather than showing them side by side — so (x + x)^2 reads as 4x^2, not 4x^1x^1. Different variable letters (e.g. x and y) stay separate, producing mixed terms like x^3y.

Privacy

This calculator runs entirely in your browser. Your terms and exponent are never uploaded, logged, or stored — the computation happens locally on your device.

Related tools

For a related two-term equation solved by a different algebraic route, see the Completing the Square Calculator.