Number of rational terms in the expansion of ${\left( {{3^{\frac{1}{8}}} + {5^{\frac{1}{3}}}} \right)^{400}}$ is
$17$
$20$
$102$
$150$
The value of $x$ in the expression ${[x + {x^{{{\log }_{10}}}}^{(x)}]^5}$, if the third term in the expansion is $10,00,000$
If the coefficient of $x$ in the expansion of ${\left( {{x^2} + \frac{k}{x}} \right)^5}$ is $270$, then $k =$
In the expansion of $(1 + x)^{43}$ if the co-efficients of the $(2r + 1)^{th}$ and the $(r + 2)^{th}$ terms are equal, the value of $r$ is :
If the coefficients of second, third and fourth term in the expansion of ${(1 + x)^{2n}}$ are in $A.P.$, then $2{n^2} - 9n + 7$ is equal to
The coefficients of three consecutive terms in the expansion of $(1+a)^{n}$ are in the ratio $1: 7: 42 .$ Find $n$