Binomial Expansion (a + b)ⁿ
a
b
n (power, max 15)
The binomial theorem describes how to expand expressions of the form (a + b)ⁿ. It uses Pascal's triangle coefficients and is fundamental to algebra, probability, and combinatorics.
- 1(a + b)ⁿ = Σ C(n,k) × aⁿ⁻ᵏ × bᵏ
- 2Coefficients C(n,k) = n! / (k!(n−k)!)
- 3Coefficients form Pascal's triangle
- 4(a+b)² = a² + 2ab + b²
(a+b)³=a³ + 3a²b + 3ab² + b³
(x+1)⁴=x⁴ + 4x³ + 6x² + 4x + 1
| n | Coefficients |
|---|---|
| 0 | 1 |
| 1 | 1 1 |
| 2 | 1 2 1 |
| 3 | 1 3 3 1 |
| 4 | 1 4 6 4 1 |
| 5 | 1 5 10 10 5 1 |
References
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