(a+b)^n = Σ[k=0 to n] nCk × a^(n-k) × b^k
Example①
Expand (x+2)^4: x⁴ + 8x³ + 24x² + 32x + 16
Expand (x+2)^4: x⁴ + 8x³ + 24x² + 32x + 16
Example② (Specific Term)
Coefficient of x³ in (2x−1)^6: when k=3, 6C3 × 2³ × (−1)³ = 20×8×(−1) = −160
Coefficient of x³ in (2x−1)^6: when k=3, 6C3 × 2³ × (−1)³ = 20×8×(−1) = −160
Key Points
- General term: nCk × a^(n-k) × b^k
- Substitute a=b=1: Σ nCk = 2^n
- Pascal's triangle: nCr = (n-1)C(r-1) + (n-1)Cr
Practice Problems
- Expand (x+1)^5.
- Find the coefficient of a²b² in (a−2b)^4.
- Find the coefficient of x^4 in (1+x)^10.
Answers
- x⁵+5x⁴+10x³+10x²+5x+1
- 4C2×(−2)²=6×4=24
- 10C4=210