Binomial Expansion Calculator – Expand (a+b)^n
Expand binomial expressions using the binomial theorem with our free online calculator. Get step-by-step expansion of (a+b)^n with binomial coefficients shown.
Understanding Binomial Expansion
The binomial expansion lets you multiply out expressions like (a + b)^n without doing all the algebra by hand. It's one of those tools that shows up everywhere — from probability to calculus to physics.
How the Binomial Theorem Works
The formula behind this calculator is:
(a + b)^n = Σ(k=0 to n) C(n,k) × a^(n-k) × b^kBreaking that down: C(n,k) is the binomial coefficient — the number of ways to choose k items from n. You'll also see it written as "n choose k" or nCk. Each term in the expansion picks a different value of k, starting from 0 and going up to n.
Worked Examples
Example 1: Expanding (x + 1)³
This is the classic introductory example. With n=3, we get 4 terms.
(x + 1)³ = x³ + 3x² + 3x + 1The coefficients 1, 3, 3, 1 come from row 3 of Pascal's triangle.
Example 2: Expanding (x + y)⁴
Now with two variables, the powers split between them.
(x + y)⁴ = x⁴ + 4x³y + 6x²y² + 4xy³ + y⁴Notice how the powers of x decrease (4→3→2→1→0) while powers of y increase (0→1→2→3→4). The coefficients 1, 4, 6, 4, 1 are symmetric.
Example 3: Expanding (2x - 3)⁵
When one term has a coefficient or is negative, you carry it through each term.
(2x - 3)⁵ = 32x⁵ - 240x⁴ + 360x³ - 270x² + 135x - 27The alternating signs come from the negative term. The coefficients get multiplied by powers of 2 and 3.
A Quick Fact
The binomial theorem for positive integer exponents was known to mathematicians in the 10th century, but Blaise Pascal's 1653 treatise on the arithmetic triangle made it widely known in Europe. Isaac Newton later generalized it to work with fractional and negative exponents — a key step in developing calculus.
Common Questions
What does the binomial coefficient represent?
C(n,k) counts the number of ways to pick k items from a set of n. In the expansion, it tells you how many different ways you can grab "a" exactly (n-k) times and "b" exactly k times when multiplying out (a+b)×(a+b)×...×(a+b).
Why do the coefficients form Pascal's triangle?
Each row of Pascal's triangle gives you the coefficients for a specific power. Row 0 is (a+b)⁰, row 1 is (a+b)¹, and so on. The pattern emerges because C(n,k) = C(n-1,k-1) + C(n-1,k) — the same rule that builds Pascal's triangle.
Can this calculator handle negative or fractional powers?
This tool handles non-negative integer exponents. For negative or fractional powers, you'd need the generalized binomial theorem, which produces infinite series instead of finite expansions.
What's the largest n I can use?
The calculator supports n up to 25. Beyond that, the coefficients get extremely large and the expansion becomes unwieldy to display. For reference, (a+b)^25 has 26 terms and the middle coefficient is over 26 billion.
How do I use this for probability?
The binomial expansion is the foundation of the binomial distribution. If you have a coin with probability p of heads, the expansion of (p + (1-p))^n gives you the probabilities of getting 0, 1, 2, ..., n heads in n flips.
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