Binomial Expansion Calculator
Expand any binomial (a + b)^n term by term and read off every binomial coefficient.
- 1
Base sum a + b
1 + 2 = 3 - 2
Number of terms n + 1
4 + 1 = 5 - 3
Total (a + b)ⁿ — sum of all binomial terms
81The binomial theorem guarantees this equals the sum of all C(n,k)·aⁿ⁻ᵏ·bᵏ terms.
How does this calculator work?
The binomial theorem expands (a + b)^n into n+1 terms, each equal to C(n,k)·a^(n-k)·b^k for k = 0…n. The coefficients C(n,k) = n!/(k!(n-k)!) are Pascal's-triangle entries. This tool lists every term, its coefficient, and its value, and verifies that the terms sum to (a + b)^n.
Formula
How this is calculated
The binomial theorem expands the power of a sum (a + b)^n into a sum of n+1 terms. The k-th term (for k = 0, 1, …, n) is C(n,k)·a^(n-k)·b^k, where the binomial coefficient C(n,k) = n!/(k!(n-k)!) counts how many ways k items can be chosen from n. Enter the first term a, the second term b, and the exponent n.
The coefficients C(n,k) form the n-th row of Pascal's triangle and are symmetric: C(n,k) = C(n,n-k). They are computed here with the multiplicative formula C(n,k) = Π[i=1..k] (n-k+i)/i, which avoids forming large factorials directly and keeps the integer coefficients exact. Each output term shows its coefficient, its symbolic form, and its numeric value for the supplied a and b; the horizontal bars compare the magnitude |value| of each term.
The exponent must be a non-negative integer (here limited to 0–60 to keep numbers finite). The sum of all numeric terms equals (a + b)^n exactly, which serves as a built-in check. Terms can be negative when a or b is negative, so the bars use absolute value while the displayed label keeps the signed amount.
Frequently asked questions
It states that (a + b)^n = Σ C(n,k)·a^(n-k)·b^k for k from 0 to n. It turns a power of a two-term sum into an explicit sum of n+1 terms whose coefficients are the binomial coefficients.
The coefficients C(n,0), C(n,1), …, C(n,n) are exactly the entries in row n of Pascal's triangle. Each entry is the sum of the two entries above it, and the row is symmetric.
The finite binomial theorem used here requires n to be a non-negative integer. Negative or fractional exponents give an infinite series instead of a finite expansion, which this calculator does not produce.
Also known as
TG we-Calculate Editorial Team. (2026). Binomial Expansion Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/binomial-expansion-calculator
TG we-Calculate Editorial Team. "Binomial Expansion Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/binomial-expansion-calculator.
TG we-Calculate Editorial Team, "Binomial Expansion Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/binomial-expansion-calculator
@misc{wecalculate_binomial_expansion_calculator, title = {Binomial Expansion Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/binomial-expansion-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?