What is the coefficient of x^{10} in the expansion of (1-x^{2})^{20}(2-x^{2}-\frac{1}{x^{2}})^{-5}?

  1. A. -1 ✓
  2. B. 1
  3. C. 10
  4. D. Coefficient of x^{10} does not exist

Correct Answer: A. -1

Explanation

(2 - x^2 - \frac{1}{x^2})^{-5} = \left(-\frac{(1-x^2)^2}{x^2}\right)^{-5} = -x^{10}(1-x^2)^{-10}. Multiplying by (1-x^2)^{20}, the given expression reduces to -x^{10}(1-x^2)^{20-10} = -x^{10}(1-x^2)^{10}. To find the coefficient of x^{10}, we need the constant term of (1-x^2)^{10}, which is 1. Multiplying by the -1 outside gives -1.

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