Sum of co-efficients of terms of degree $m$  in the expansion of $(1 + x)^n(1 + y)^n(1 + z)^n$ is

Given that $4^{th}$ term in the expansion of ${\left( {2 + \frac{3}{8}x} \right)^{10}}$ has the maximum numerical value, the range of value of $x$ for which this will be true is given by

The coefficient of $x^{4}$ is the expansion of $\left(1+\mathrm{x}+\mathrm{x}^{2}\right)^{10}$ is

The term independent of $' x '$ in the expansion of ${\left( {9\,x\,\, - \,\,\frac{1}{{3\,\sqrt x }}} \right)^{18}}, x > 0$ , is $\alpha$ times the corresponding binomial co-efficient . Then $' \alpha '$ is :

If the coefficients of ${r^{th}}$ term and ${(r + 4)^{th}}$ term are equal in the expansion of ${(1 + x)^{20}}$, then the value of r will be

Find the coefficient of $a^{4}$ in the product $(1+2 a)^{4}(2-a)^{5}$ using binomial theorem.