The coefficient of $x^4$ in the expansion of $(1-x+x^2-x^3)^4$ is

The coefficient of $x^{-6}$ in the expansion of $\left(\frac{4x}{5} + \frac{5}{2x^2}\right)^9$ is $........$.

If $n$ is an even positive integer,then the condition that the greatest term in the expansion of $(1 + x)^n$ may have the greatest coefficient also,is

The coefficient of $x^{n-6}$ in the expansion $n! \left[ x - \left( \frac{^nC_0 + ^nC_1}{^nC_0} \right) \right] \left[ \frac{x}{2} - \left( \frac{^nC_1 + ^nC_2}{^nC_1} \right) \right] \left[ \frac{x}{3} - \left( \frac{^nC_2 + ^nC_3}{^nC_2} \right) \right] \dots \left[ \frac{x}{n} - \left( \frac{^nC_{n-1} + ^nC_n}{^nC_{n-1}} \right) \right]$ is equal to:

The sum of the coefficients of the first $50$ terms in the binomial expansion of $(1-x)^{100}$ is equal to

Let $(x + 10)^{50} + (x - 10)^{50} = a_0 + a_1x + a_2x^2 + .... + a_{50}x^{50}$,for $x \in R$; then $\frac{a_2}{a_0}$ is equal to