Binomial Expansion Calculator to Expand (a + b)^n via Pascal's Triangle

The Binomial Theorem

The Binomial Theorem provides a formula for expanding powers of a binomial expression (a + b)^n without repeated multiplication. Each term uses a binomial coefficient (C(n,k)) which corresponds directly to entries in Pascal's Triangle. It is used in algebra, probability, statistics, and combinatorics.

🔢 Binomial Expansion Calculator

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📐 Formula

(a+b)^n = Σ C(n,k) × a^(n-k) × b^k

C(n,k) = n! / (k!(n-k)!) is the binomial coefficient (also written ⁿCₖ or "n choose k"). k ranges from 0 to n.

📝 Worked Example

Expand (x+2)³ (n=3, a=x, b=2):
= C(3,0)x³2⁰ + C(3,1)x²2¹ + C(3,2)x¹2² + C(3,3)x⁰2³
= 1x³ + 3x²(2) + 3x(4) + 8
= x³ + 6x² + 12x + 8

📝 How to Use the Calculator

Enter BinomialInput the two terms a and b (numbers, variables, or expressions).

Enter Power nThe exponent n (integer ≥ 0).

ExpandView the full expanded polynomial with all terms.

Find Specific TermEnter k to find the (k+1)th term without expanding everything.

❓ FAQ

What is Pascal's Triangle?

A triangular array of binomial coefficients. Each row n gives the coefficients for (a+b)^n. Each number is the sum of the two numbers above it.

How is the Binomial Theorem used in probability?

In a Bernoulli process (n independent trials, probability p of success), P(exactly k successes) = C(n,k) × p^k × (1-p)^(n-k) — the binomial distribution.

The Binomial Theorem