For the natural numbers $m, n$, if $(1-y)^{m}(1+y)^{n}=1+a_{1} y+a_{2} y^{2}+\ldots .+a_{m+n} y^{m+n}$ and $a_{1}=a_{2}$ $=10$, then the value of $(m+n)$ is equal to:

Let the coefficients of third, fourth and fifth terms in the expansion of $\left(x+\frac{a}{x^{2}}\right)^{n}, x \neq 0,$ be in the ratio $12: 8: 3 .$ Then the term independent of $x$ in the expansion, is equal to ...... .

Middle term in the expansion of ${(1 + 3x + 3{x^2} + {x^3})^6}$ is

If the coefficients of ${x^2}$ and ${x^3}$ in the expansion of ${(3 + ax)^9}$ are the same, then the value of $a$ is

The number of rational terms in the binomial expansion of $\left(4^{\frac{1}{4}}+5^{\frac{1}{6}}\right)^{120}$ is $....$

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