Binomial Expansion Calculator

(a+b)^n coefficients · Pascal triangle

Quick Answer

Binomial Expansion Calculator: (a+b)^n coefficients · Pascal triangle. Binomial Expansion Calculator uses standard math formulas to compute accurate results instantly.

Exponent n (0-12)

Query term k (0-based)

(a+b)^5 =

a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5

Coefficients6

Coefficient sum2^5 = 32

Max coefficient10

Expansion steps

1(a+b)^5 = Σ C(5,k)·a^(5−k)·b^k
2Pascal row 5: 1, 5, 10, 10, 5, 1
3Sum of coefficients = 2^5 = 32; sum of odd-position = even-position = 2^4 = 16
4Term 3: C(5,2)·a^3·b^2 = 10·a^3·b^2

Pascal's triangle (rows 0-5)

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

The binomial theorem: (a+b)ⁿ = Σ C(n,k)aⁿ⁻ᵏbᵏ. Each coefficient row is Pascal's triangle — every entry is the sum of the two above it. Setting a=b=1 proves the row sum is 2ⁿ; setting a=1, b=−1 proves odd and even positions have equal sums. Key applications: binomial probability, series approximations like (1+x)ⁿ ≈ 1+nx for small x (the basis of compound-interest shortcuts).

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Key capability: (a+b)^n coefficients · Pascal triangle.
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Formula & How It Works

Binomial Expansion Calculator formula

Binomial Expansion Calculator uses standard math formulas to compute accurate results instantly.

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Source & Updates

Standard mathematical formulas. All logic verified. Last updated: 2025-07

Last updated: 2026-08-05

Formulas and content on this page are reviewed by the YiJianSuan team. For reference only — official sources prevail.

Cite this page: YiJianSuan. Binomial Expansion Calculator[EB/OL]. https://www.yijiansuan.com/en/c/math/binomial-expansion

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Pascal's triangle (rows 0-5)

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Pascal's triangle (rows 0-5)