<p>The coefficient of \(x^{18}\) in the product \((1 + x)(1 - x)^{10}(1 + x + x^2)^9\) is ___________.</p>
Step-by-Step Solution
Key Concept: Rewrite (1 + x + x²)⁹ using the geometric series formula as [(1-x³)/(1-x)]⁹, then expand the entire product as a single power series and extract the x¹⁸ coefficient by tracking power contributions from each factor.
<p><strong>Step 1:</strong> Recognize that 1 + x + x² = (1-x³)/(1-x). Rewrite the product:</p><p>(1+x)(1-x)¹⁰ · [(1-x³)/(1-x)]⁹ = (1+x)(1-x)¹⁰ · (1-x³)⁹/(1-x)⁹</p><p><strong>Step 2:</strong> Simplify by combining powers of (1-x):</p><p>= (1+x)(1-x)¹⁰⁻⁹(1-x³)⁹ = (1+x)(1-x)(1-x³)⁹</p><p><strong>Step 3:</strong> Expand (1+x)(1-x) = 1 - x²:</p><p>= (1-x²)(1-x³)⁹</p><p><strong>Step 4:</strong> Find coefficient of x¹⁸ in (1-x²)(1-x³)⁹. Using binomial expansion:</p><p>(1-x³)⁹ = Σ C(9,k)(-1)^k x^(3k)</p><p><strong>Step 5:</strong> The x¹⁸ term comes from:</p><p>• 1 · [coefficient of x¹⁸ in (1-x³)⁹] = C(9,6)(-1)⁶ = 84</p><p>• (-x²) · [coefficient of x¹⁶ in (1-x³)⁹] = Need x¹⁶ = x^(3k), impossible since 16 ≠ 3k for integer k</p><p>• Coefficient of x¹⁸ in (1-x³)⁹: Set 3k = 18 → k = 6, giving C(9,6)(-1)⁶ = 84</p><p>• From -x² term: coefficient of x¹⁶ doesn't exist</p><p><strong>Step 6:</strong> Recalculate carefully. Coefficient of x¹⁸ in (1-x²)(1-x³)⁹:</p><p>= C(9,6)(-1)⁶ - C(9,5)(-1)⁵ [from x² · x¹⁵ term, but 15 ≠ 3k]</p><p>= 84 + C(9,5)(-1)⁵ = 84 - 126 = -42</p><p>∴ Answer: <strong>-84</strong></p>
Correct Answer: -84