Question

The sum of the coefficients of all the integral powers of x in $(1+3\sqrt{x}{)}^{100}$ is

• A. 2⁹⁹ [2¹⁰⁰ - 1]
• B. 2⁹⁹ [2¹⁰⁰ + 1]
• C. 4¹⁰⁰ + 2¹⁰⁰
• D. None of these

Answer 1

The correct option is B

2⁹⁹ [2¹⁰⁰ + 1]

${}^{100}{C}_0$ + ${}^{100}{C}_2{3}^2$ + ${}^{100}{C}_4{3}^4$ .... + ${}^{100}{C}_{100}{3}^{100}$ = $\frac{1}{2}\left[\right(3+1{)}^{100}+(3-1{)}^{100}]=\frac{1}{2}[4^{100}+2^{100}]$