Binomial Expansion Calculator — Formula
## 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
### 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
Esempio Risolto
Find the k=2 term of (2+3)^4.
- C(4,2) = 6
- a^(4-2) = 2² = 4
- b^2 = 3² = 9
- Term value = 6 × 4 × 9 = 216
- (2+3)⁴ = 5⁴ = 625