<p>Given <br>\((1+x)(1-x)^{10}(1+x+x^2)^9\)<br>Find the coefficient of \(x^{18}\) in the expansion.</p>
Step-by-Step Solution
Key Concept: Rewrite (1+x+x²)⁹ using the identity 1+x+x² = (1-x³)/(1-x), then expand (1+x)(1-x)¹⁰(1-x³)⁹/(1-x)¹⁰ = (1+x)(1-x³)⁹(1-x)⁻⁹ to find the coefficient of x¹⁸ using binomial series.
<p><strong>Step 1:</strong> Use the identity 1+x+x² = (1-x³)/(1-x)</p><p>(1+x)(1-x)¹⁰(1+x+x²)⁹ = (1+x)(1-x)¹⁰ · [(1-x³)/(1-x)]⁹</p><p>= (1+x)(1-x)¹⁰ · (1-x³)⁹(1-x)⁻⁹</p><p>= (1+x)(1-x)·(1-x³)⁹</p><p><strong>Step 2:</strong> Simplify (1+x)(1-x) = 1-x²</p><p>(1-x²)(1-x³)⁹</p><p><strong>Step 3:</strong> Expand (1-x³)⁹ using binomial theorem:</p><p>(1-x³)⁹ = Σ C(9,r)(-1)ʳ(x³)ʳ = Σ C(9,r)(-1)ʳx³ʳ</p><p><strong>Step 4:</strong> Find coefficient of x¹⁸ in (1-x²)·(1-x³)⁹</p><p>From (1-x²)(1-x³)⁹, we need:</p><p>• Coefficient of x¹⁸ from 1·(1-x³)⁹: This comes from C(9,6)(-1)⁶ = C(9,6) = 84 (when 3r=18, r=6)</p><p>• Coefficient of x¹⁶ from -x²·(1-x³)⁹: This comes from -C(9,r)(-1)ʳ where 3r+2=18, giving r=16/3 (not integer, so 0)</p><p>∴ Answer: <strong>84</strong></p>
Correct Answer: 84