Question

The sum of coefficients of intergral power of $x$ in the binomial expansion ${(1-2\sqrt{x})}^{50}$is:

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

Answer 1

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

$(1+2\sqrt{x}{)}^{50}={}^{50}{C}_0+{}^{50}{C}_{1}2\sqrt{x}+{}^{50}{C}_2(2\sqrt{x}{)}^2...\left(1\right)(1-2\sqrt{x}{)}^{50}={}^{50}{C}_0+{}^{50}{C}_{1}2\sqrt{x}-{}^{50}{C}_2(2\sqrt{x}{)}^2+{}^{50}{C}_3(2\sqrt{x}{)}^3-{}^{50}{C}_4(2\sqrt{x}{)}^4...\left(2\right)\text{Adding equation (1)and (2)}(1+2\sqrt{x}{)}^{50}+(1-2\sqrt{x}{)}^{50}=2{[}^{50}{C}_0+{}^{50}{C}_2{2}^{2}x{+}^{50}{C}_4{2}^3{x}^2+...]\text{Putting}x=1,\text{we get above as}\frac{3^{50}+1}{2}$