The coefficient of $x^{101}$ in the expression $(5+x)^{500} + x(5+x)^{499} + x^{2}(5+x)^{498} + \ldots + x^{500}$ for $x > 0$ is:

Out of $5$ apples,$10$ mangoes,and $15$ oranges,$15$ fruits are to be distributed between two persons. Find the total number of ways of distribution.

Using the Binomial Theorem,evaluate $(96)^{3}$.

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $(x+\sqrt{x^3-1})^5+(x-\sqrt{x^3-1})^5, x>1$. If $u$ and $v$ satisfy the equations $\alpha u+\beta v=18$ and $\gamma u+\delta v=20$,then $u+v$ equals:

If the ratio of the coefficients of the third and fourth terms in the expansion of $(x - \frac{1}{2x})^n$ is $1 : 2$,then the value of $n$ is:

In the expansion of $(a+1+\frac{1}{a})^n$,where $n \in N$,there are $2029$ terms. Then $n=$