Question Detail
Question
Find the coefficient of \( x^{-12} \) in \( (x^3 - 1/x)^{24} \).
Correct Answer
Option A is the correct answer.
Detailed Explanation
The general term is T_{r+1} = C(24, r) (x^3)^{24-r} (-1/x)^r = C(24, r) x^{72-3r} * (-1)^r * x^{-r} = C(24, r) (-1)^r x^{72-4r}. We need 72 - 4r = -12 => 4r = 84 => r = 21. So the coefficient is C(24,21) * (-1)^{21} = C(24,3) * (-1) = 2024 * (-1) = -2024. However, the options include -2024 (B) and -134596 (A). Recalculating: C(24,3) = 242322/6 = 2024. So -2024. There is a discrepancy: option B is -2024, which is correct. Option A is -134596, which is much larger. Possibly the exponent in the binomial was different. Given the calculation, the correct answer is -2024.
Hint
Use the general term formula and set the exponent of x to the required value.