Binomial expansion calculator
Expand a numeric binomial (a + b)^n term by term.
What this calculator covers
Use this binomial expansion calculator to expand a numeric binomial of the form (a + b)^n.
Each term is generated from the binomial coefficients, so you can see how the theorem builds the final total instead of treating the result like a black box.
Frequently asked questions
- What is a binomial coefficient?
- A binomial coefficient C(n, k) counts how many ways you can choose k items from a set of n. In the expansion, it weights each term to account for the number of identical combinations that contribute to that term's total.
- Does this calculator handle symbolic variables like x and y?
- No. The inputs are numeric values for a and b, so the calculator evaluates each term directly and sums to a single number. For symbolic expansion like (x + 2)^4, you would need an algebra system.
- What is the maximum exponent supported?
- The exponent must be a whole number from 0 through 20. Larger exponents produce very large intermediate coefficients that can exceed reliable floating-point precision.
- Why does (a + b)^0 equal 1?
- Any non-zero number raised to the power of zero equals 1 by convention. With n = 0 there is exactly one term — C(0, 0) × a^0 × b^0 = 1 — so the expansion always returns 1 regardless of the values of a and b.
Tool
Run the calculation
Result
RESULT · BINOMIAL
â„–212
Primary result
81 + 108 + 54 + 12 + 1 = 256
Using the binomial theorem, (3 + 1)^4 expands to 81 + 108 + 54 + 12 + 1 = 256.
- Expanded terms
- 81, 108, 54, 12, 1
- Expansion
- 81 + 108 + 54 + 12 + 1 = 256
- Term count
- 5
- Total
- 256
Step-by-step solution
- 1.Use the coefficients C(4, k) for k = 0 through 4 to weight each term of the expansion.
- 2.Compute each term as C(4, k) · 3^(4 - k) · 1^k.
- 3.Add the term values together to get 256.
Walkthrough
Visual walkthrough
The binomial theorem builds the result by pairing each power of the first term with the matching power left over for the second term.
01
Read the exponent ladder
(3 + 1)^4
The first term starts with the full exponent and the second term gains one power at each step.
02
Apply the binomial coefficients
C(4, 0), C(4, 1), C(4, 2), C(4, 3), C(4, 4)
Those coefficients count how many identical term combinations appear in the full expansion.
03
Add the evaluated terms
Once every weighted term is evaluated numerically, their sum is the expanded result.
81 + 108 + 54 + 12 + 1 = 256