Binomial Theorem Calculator – Expand (a+b)ⁿ

The binomial theorem expands a power of a two-term sum into a polynomial whose coefficients are the combinations C(n,k). Expanding (x+2)⁵ pulls coefficients from Pascal’s triangle row 1, 5, 10, 10, 5, 1: the result is x⁵ + 5·2x⁴ + 10·4x³ + 10·8x² + 5·16x + 32, which simplifies to x⁵ + 10x⁴ + 40x³ + 80x² + 80x + 32. The calculator prints the raw term C(5,k)·xⁿ⁻ᵏ·2ᵏ before simplification so the structure is visible, plus a specific-term finder: the x² coefficient of (x+3)⁴ is C(4,2)·3² = 6×9 = 54 without expanding anything. Special cases fall out instantly — (a+b)² = a²+2ab+b² and (1+x)⁶ has coefficients 1, 6, 15, 20, 15, 6, 1. The symmetric coefficient pattern and the sum of all coefficients, 2ⁿ = 32 for n = 5, are shown as built-in sanity checks.

1. 1

   Enter a, b and n

   For example x, 2 and 5 to expand (x+2)⁵

2. 2

   Fetch the coefficients

   Pascal’s triangle row n or C(n,k) values are listed

3. 3

   Build each term

   C(n,k)·aⁿ⁻ᵏ·bᵏ terms are shown before simplification

4. 4

   Sum and verify

   Coefficients total 2ⁿ; substitute a=b=1 as a numeric check

• 1

  Pascal’s triangle row n lists exactly the C(n,k) coefficients

• 2

  The k-th term is C(n,k)·aⁿ⁻ᵏ·bᵏ — no expansion needed

• 3

  Sum of coefficients = (1+1)ⁿ = 2ⁿ

• Using permutations nPr instead of combinations C(n,k) for coefficients

• Forgetting the bᵏ power on the second term of each coefficient pair

• Sign slips with (a−b)ⁿ — odd k terms flip sign

• Stopping one term early; the expansion has n+1 terms