If ${x^4}$ occurs in the ${r^{th}}$ term in the expansion of ${\left( {{x^4} + \frac{1}{{{x^3}}}} \right)^{15}}$, then $r = $

If the coefficient of $4^{th}$ term in the expansion of ${(a + b)^n}$ is $56$, then $n$ is

Find the term independent of $x$ in the expansion of $\left(\frac{3}{2} x^{2}-\frac{1}{3 x}\right)^{6}$

The middle term in the expansion of ${\left( {x + \frac{1}{x}} \right)^{10}}$ is

${r^{th}}$ term in the expansion of ${(a + 2x)^n}$ 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