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

If the coefficients of $x^2$ and $x^3$ are both zero, in the expansion of the expression $(1 + ax + bx^2) (1 -3x)^{t5}$ in powers of $x$, then the ordered pair $(a, b)$ is equal to

Write the general term in the expansion of $\left(x^{2}-y\right)^{6}$

Let $a$ and $b$ be two nonzero real numbers. If the coefficient of $x^5$ in the expansion of $\left(a x^2+\frac{70}{27 b x}\right)^4$ is equal to the coefficient of $x^{-5}$ is equal to the coefficient of $\left(a x-\frac{1}{b x^2}\right)^7$, then the value of $2 b$ is

The positive value of $\lambda $ for which the co-efficient of $x^2$ in the expression ${x^2}{\left( {\sqrt x  + \frac{\lambda }{{{x^2}}}} \right)^{10}}$ is $720$ is

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