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The sum of the coefficients of all the integral powers of x in the expansion of $(1 + 2 \sqrt{x})^{80}$ is

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a

12380+1

b

12380−1

c

380+1

d

380-1

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detailed solution

Correct option is A

The coefficients of the integral powers of x are 80C0,80C2⋅22,80C4⋅24,…,80C80⋅280Now, (1+2)80=80C0+80C1⋅2+80C2⋅22+…+80C80⋅280 ……. (1)and (1−2)80=80C0−80C12+80C2⋅22−…+80C80⋅280 …….. (2)Adding Eq. (1) and (2), we get380+1=2 80C0+80C2⋅22+80C4⋅24+…+80C80⋅280∴ 80C0+80C2⋅22+80C4⋅24+…+80C80⋅280=12380+1

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