If the coefficients of $x$ and $x^{2}$ in the expansion of $(1+x)^{p}(1-x)^{q}, p, q \leq 15$, are $-3$ and $-5$ respectively, then the coefficient of $x ^{3}$ is equal to $............$

The term independent of $x$ in the expansion of ${\left( {\frac{1}{2}{x^{1/3}} + {x^{ - 1/5}}} \right)^8}$ will be

Let the sixth term in the binomial expansion of $\left(\sqrt{2^{\log _2}\left(10-3^x\right)}+\sqrt[5]{2^{(x-2) \log _2 3}}\right)^m$, in the increasing powers of $2^{(x-2) \log _2 3}$, be $21$ . If the binomial coefficients of the second, third and fourth terms in the expansion are respectively the first, third and fifth terms of an $A.P.$, then the sum of the squares of all possible values of $x$ is $.........$.

The term independent of $x$ in the expansion ${\left( {{x^2} - \frac{1}{{3x}}} \right)^9}$ is

The ratio of the coefficient of terms ${x^{n - r}}{a^r}$and ${x^r}{a^{n - r}}$ in the binomial expansion of ${(x + a)^n}$ will be

For a positive integer $n,\left(1+\frac{1}{x}\right)^{n}$ is expanded in increasing powers of $x$. If three consecutive coefficients in this expansion are in the ratio, $2: 5: 12,$ then $n$ is equal to