The coefficient of $x ^7$ in $\left(1-x+2 x^3\right)^{10}$ is $........$.

If the sum of the coefficients in the expansion of ${(x + y)^n}$ is $1024$, then the value of the greatest coefficient in the expansion is

If the ratio of the coefficient of third and fourth term in the expansion of ${\left( {x - \frac{1}{{2x}}} \right)^n}$ is $1 : 2$, then the value of $ n$ will be

Let $[t]$ denotes the greatest integer $\leq t$. If the constant term in the expansion of $\left(3 x^2-\frac{1}{2 x^5}\right)^7$ is $\alpha$, then $[\alpha]$ is equal to $............$.

In the binomial expansion of ${\left( {a - b} \right)^n},n \ge 5,\;$ the sum of $5^{th}$ and $6^{th}$ terms is zero , then $a/b$ equals.

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