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Binomial Expansion Calculator

Math Calculators

(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.

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

Pascal's triangle (rows 0-5)

   1
   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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Formula & How It Works

Binomial Expansion Calculator formula

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

FAQ

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Source & Updates

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

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