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The coefficient of x²⁰ in the expansion of ${(1 + x^{2})}^{40} \cdot {(x^{2} + 2 + \frac{1}{x^{2}})}^{- 5}$ is

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a

30C10

b

30C25

c

1

d

None of these

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detailed solution

Correct option is B

Given,1+x240⋅x2+2+1x2−5=1+x240x+1x2−5=1+x240x+1x−10=x101+x2401+x2−10=x101+x230to find The coefficient of x20 in the expansion of1+x220x+1x−10 i.e. x101+x230Now, 1+x230=30C0+30C1x2+30C2x22+30C3x23+30C4x24+⋯+30C30x230… (i) x101+x230=x10+30C1x12+30C2x14+30C3x16+30C4x18+30C5x20+…+x70∴ Coefficient of x20 is 30C5 or 30C25. ∵nCr=nCn−r

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