What is the binomial theorem?

The binomial theorem provides a formula for expanding (a+b)^n as a sum of terms involving powers of a and b, weighted by binomial coefficients.

How many terms in a binomial expansion?

(a+b)^n has exactly n+1 terms, corresponding to k = 0, 1, 2, ..., n.

What is the middle term?

When n is even, the middle term is at k = n/2. When n is odd, there are two middle terms at k = (n-1)/2 and k = (n+1)/2.

Binomial Expansion Calculator

Calculate individual terms of the binomial expansion (a + b)^n. Shows the coefficient, power of a, and power of b for a specific term.

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Value of a

Value of b

Exponent n

Term k (0-indexed)

Total

625.0000

Coeff6

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Binomial Expansion

Binomial Theorem

(a + b)^n = sum of C(n,k) × a^(n-k) × b^k for k = 0 to n

Individual Term

The k-th term (0-indexed): C(n,k) × a^(n-k) × b^k

Example: (2 + 3)⁴ = 5⁴ = 625

k=0: C(4,0)×2⁴×3⁰ = 1×16×1 = 16

k=1: C(4,1)×2³×3¹ = 4×8×3 = 96

k=2: C(4,2)×2²×3² = 6×4×9 = 216

k=3: C(4,3)×2¹×3³ = 4×2×27 = 216

k=4: C(4,4)×2⁰×3⁴ = 1×1×81 = 81

Total: 16 + 96 + 216 + 216 + 81 = 625

Example Calculation

Find the k=2 term of (2+3)^4.

01C(4,2) = 6
02a^(4-2) = 2² = 4
03b^2 = 3² = 9
04Term value = 6 × 4 × 9 = 216
05(2+3)⁴ = 5⁴ = 625

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Binomial Expansion Calculator

Binomial Expansion

Binomial Theorem

Individual Term

Example: (2 + 3)⁴ = 5⁴ = 625

Example Calculation

Frequently Asked Questions

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