Using the Binomial Theorem,evaluate $(102)^{5}$.

If $x$ is so small that $x^3$ and higher powers of $x$ may be neglected,then $\frac{(1 + x)^{3/2} - (1 + \frac{1}{2}x)^3}{(1 - x)^{1/2}}$ may be approximated as

When $|x| < 2$,the coefficient of $x^2$ in the power series expansion of $\frac{x}{(x-2)(x-3)}$ is

The last digit in $7^{300}$ is

Let $n > 2$ be an integer and define a polynomial $p(x) = x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$,where $a_0, a_1, \ldots, a_{n-1}$ are integers. Suppose we know that $n p(x) = (1 + x) p'(x)$. If $b = p(1)$,then:

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.