How Pascal's triangle coefficients expand (a + b)ⁿ to any degree without tedious algebra.
Expanding Binomials Effortlessly
Multiplying (a + b)⁵ by hand requires tedious FOIL expansion. The Binomial Theorem provides the exact algebraic formula in one step.
Binomial Formula: (a + b)ⁿ = Σ [ nCr · aⁿ⁻ʳ · bʳ ] from r = 0 to n
Pascal's Triangle Pattern
Each entry is the direct sum of the two numbers directly above it. The n-th row gives the exact polynomial coefficients for (a + b)ⁿ.
Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1
Row 5: 1 5 10 10 5 1
Worked Example: Expand (x + 2)⁴
Using Row 4 coefficients (1, 4, 6, 4, 1): (x + 2)⁴ = 1(x⁴) + 4(x³)(2) + 6(x²)(2²) + 4(x)(2³) + 1(2⁴).
Simplified: x⁴ + 8x³ + 24x² + 32x + 16
Hidden Patterns in Pascal's Triangle
Row sums equal powers of 2 (2ⁿ). Diagonals reveal triangular numbers (1, 3, 6, 10, 15). Coloring odd numbers generates the Sierpinski Triangle fractal.
Row Sum Identity: Σ nCr = 2ⁿ
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