Single Choice

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

A$$\dfrac {1}{2}(3^{50}+1)$$
Correct Answer
B$$\dfrac {1}{2}(3^{50})$$
C$$\dfrac {1}{2}(3^{50}-1)$$
D$$\dfrac {1}{2}(2^{50}+1)$$

Solution

$$(1-2\sqrt{x})^{50}= ^{50}C_0-^{50}C_1(2\sqrt{x})+^{50}C_2(2\sqrt{x})^2.....$$

we need to find sum of even forms, so

Put $$\sqrt{x}=1 \Longrightarrow (1-2)^{50}=1=^{50}C_0-^{50}C_1(2)+^{50}C_2(2)^2.........(1)$$

and $$\sqrt{x}=-1 \Longrightarrow (1+2)^{50}=3^{50}=^{50}C_0+^{50}C_12+^{50}C_2(2)^2.......(2)$$

Adding (1) and (2) we get,

$$ 3^{50}+1=2(^{50}C_0+^{50}C_2(2)^2+........)$$

$$\Rightarrow (^{50}C_0+^{50}C_2(2)^2+........)=\cfrac{1}{2}(1+3^{50})$$

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