<p>In the expansion of \(\left(x^3 - \dfrac{1}{x^2}\right)^{15}\), the constant term is</p>
Step-by-Step Solution
Key Concept: In the binomial expansion of (x³ - 1/x²)¹⁵, the general term is C(15,r)(x³)^(15-r)(-1/x²)^r = C(15,r)(-1)^r·x^(45-3r-2r). For a constant term, the power of x must equal zero, so 45-5r=0.
<p><strong>Step 1:</strong> Write the general term in the expansion of (x³ - 1/x²)¹⁵</p><p>T_(r+1) = C(15,r)·(x³)^(15-r)·(-1/x²)^r = C(15,r)·(-1)^r·x^(45-3r)·x^(-2r) = C(15,r)·(-1)^r·x^(45-5r)</p><p><strong>Step 2:</strong> For constant term, power of x = 0</p><p>45 - 5r = 0 ⟹ r = 9</p><p><strong>Step 3:</strong> Calculate the constant term</p><p>T₁₀ = C(15,9)·(-1)^9 = C(15,9)·(-1)</p><p>C(15,9) = C(15,6) = 15!/(9!·6!) = (15×14×13×12×11×10)/(6×5×4×3×2×1) = 5005</p><p><strong>Step 4:</strong> Final answer</p><p>Constant term = -5005</p><p>∴ Answer: <strong>-5005</strong> (or verify with option C)</p>
Correct Answer: C