<p>Given <br>
\((1-x)^2(1+x^2)^3(1+x^3)^4\)<br>
Find the coefficient of \(x^{10}\).</p>
Step-by-Step Solution
Key Concept: Expand each factor strategically using binomial theorem, then identify all combinations of terms from each factor that multiply to give exactly x^10. Track which power from each factor contributes to the final degree.
<p><strong>Step 1:</strong> Expand using binomial theorem:</p><p>(1-x)^2 = 1 - 2x + x^2</p><p>(1+x^2)^3 = C(3,0) + C(3,1)x^2 + C(3,2)x^4 + C(3,3)x^6 = 1 + 3x^2 + 3x^4 + x^6</p><p>(1+x^3)^4 = C(4,0) + C(4,1)x^3 + C(4,2)x^6 + C(4,3)x^9 + C(4,4)x^12 = 1 + 4x^3 + 6x^6 + 4x^9 + x^12</p><p><strong>Step 2:</strong> Find all combinations where powers sum to 10.</p><p>From (1-x)^2, (1+x^2)^3, (1+x^3)^4 respectively, we need: a + b + c = 10 where a ∈ {0,1,2}, b ∈ {0,2,4,6}, c ∈ {0,3,6,9,...}</p><p><strong>Step 3:</strong> Valid combinations:</p><p>• (x^0)(x^4)(x^6): coeff = 1 × 3 × 6 = 18</p><p>• (x^0)(x^6)(x^4): Not possible (x^4 not available)</p><p>• (x^1)(x^0)(x^9): coeff = (-2) × 1 × 4 = -8</p><p>• (x^1)(x^6)(x^3): coeff = (-2) × 3 × 4 = -24</p><p>• (x^2)(x^4)(x^4): Not possible (x^4 appears once)</p><p>• (x^2)(x^0)(x^8): Not possible (x^8 not available)</p><p>• (x^0)(x^2)(x^8): Not possible</p><p>• (x^4)(x^6)(x^0): coeff = 1 × 3 × 1 = 3</p><p><strong>Step 4:</strong> Sum all coefficients: 18 + (-8) + (-24) + 3 = -11</p><p>∴ Answer: A</p>
Correct Answer: A