${r^{th}}$ term in the expansion of ${(a + 2x)^n}$ is
$\frac{{n(n + 1)....(n - r + 1)}}{{r!}}{a^{n - r + 1}}{(2x)^r}$
$\frac{{n(n - 1)....(n - r + 2)}}{{(r - 1)\,!}}{a^{n - r + 1}}{(2x)^{r - 1}}$
$\frac{{n(n + 1)....(n - r)}}{{(r + 1)!}}{a^{n - r}}{(x)^r}$
None of these
A ratio of the $5^{th}$ term from the beginning to the $5^{th}$ term from the end in the binomial expansion of $\left( {{2^{1/3}} + \frac{1}{{2{{\left( 3 \right)}^{1/3}}}}} \right)^{10}$ is
The term independent of $x$ in the expansion ${\left( {{x^2} - \frac{1}{{3x}}} \right)^9}$ is
If the third term in the binomial expansion of ${(1 + x)^m}$ is $ - \frac{1}{8}{x^2}$, then the rational value of $m$ is
The term independent of $x$ in the expansion of $\left[\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right]^{10}, x \neq 1,$ is equal to ....... .
For a positive integer $n,\left(1+\frac{1}{x}\right)^{n}$ is expanded in increasing powers of $x$. If three consecutive coefficients in this expansion are in the ratio, $2: 5: 12,$ then $n$ is equal to