Binomial Theorem Calculator

Expand any binomial (a + b)ⁿ for n up to 10. Displays each term using C(n,k)·a^(n−k)·b^k, shows the Pascal's triangle row, and accepts custom variable names.

a coefficient

a variable

b coefficient

b variable

Exponent n (0–10)

Expansion

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Terms Breakdown

k	C(n,k)	aⁿ⁻ᵏ	bᵏ	Term

Pascal's Triangle Row n

The Binomial Theorem

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

where C(n,k) = n! / (k! × (n−k)!) is the binomial coefficient.

'What is Pascal\'s Triangle?', 'answer' => 'Pascal\'s Triangle is a triangular array where each number is the sum of the two numbers above it. Each row n gives the binomial coefficients C(n,0), C(n,1), ..., C(n,n) for the expansion of (a+b)ⁿ.'], ['question' => 'How many terms does a binomial expansion have?', 'answer' => 'An expansion of (a+b)ⁿ has exactly n+1 terms (one for each k from 0 to n).'], ['question' => 'What is the middle term of a binomial expansion?', 'answer' => 'If n is even, there is one middle term at k=n/2. If n is odd, there are two middle terms at k=(n−1)/2 and k=(n+1)/2. The middle terms are often the largest when a=b=1.'], ]" />

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