# The sum of coefficients of integral powers of  x  in  the  binomial  expansion  of Option 1) Option 2) Option 3) Option 4)

As we learnt in

Properties of Binomial Theorem -

$\dpi{120} \left ( x+a \right )^{n}+\left ( x-a \right )^{n}= 2\left ( ^{n}c_{0} \, x^{n}+ ^{n}c_{2}\, x^{n-2}\, a^{2}+---\right )$

- wherein

Sum of odd terms or even Binomial coefficients

$\left(1-2\sqrt{x} \right )^{50}=1-^{50}C_{1}\ 2\sqrt{x}+^{50}C_{2}\ (2\sqrt{x})^{2}-..................$

$\underline\left(1+2\sqrt{x} \right )^{50}=1-^{50}C_{1}\ 2\sqrt{x}+^{50}C_{2}\ (2\sqrt{x})^{2}+..................$

Add and put x = 1

$1+3^{50}=2 \left(1+^{50}C_{2}\ 2x+............. \right )$

Sum of integral powers = $=\frac{3^{50}+1}{2}$

Correct option is 1.

Option 1)

This is the correct option.

Option 2)

This is an incorrect option.

Option 3)

This is an incorrect option.

Option 4)

This is an incorrect option.

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