Question

The sum of coefficient of integral powers of $x$ in the binomial expansion of $(1-2\sqrt{x}{)}^{50}$ is :

• A. $\frac{1}{2}(3^{50}+1)$
• B. $\frac{1}{2}\left(3^{50}\right)$
• C. $\frac{1}{2}(3^{50}-1)$
• D. $\frac{1}{2}(2^{50}+1)$

Answer 1

The correct option is A $\frac{1}{2}(3^{50}+1)$

$(1-2\sqrt{x}{)}^{50}{=}^{50}{C}_0{-}^{50}{C}_1(2\sqrt{x}\left){+}^{50}{C}_2\right(2\sqrt{x}{)}^2.....$

we need to find sum of even forms, so

Put $\sqrt{x}=1⟹(1-2{)}^{50}=1{=}^{50}{C}_0{-}^{50}{C}_1(2\left){+}^{50}{C}_2\right(2{)}^2.........\left(1\right)$

and $\sqrt{x}=-1⟹(1+2{)}^{50}=3^{50}{=}^{50}{C}_0{+}^{50}{C}_{1}2{+}^{50}{C}_2(2{)}^2.......\left(2\right)$

Adding (1) and (2) we get,

$3^{50}+1=2{(}^{50}{C}_0{+}^{50}{C}_2(2{)}^2+........)$

$\Rightarrow {(}^{50}{C}_0{+}^{50}{C}_2(2{)}^2+........)=\frac{1}{2}(1+3^{50})$