Using the Binomial Theorem,evaluate $(99)^{5}$.

The value of $\sum\limits_{r = 1}^{19} {\frac{{{}^{20}{C_{r + 1}}{(-1)}^r}}{{{2^{2r + 1}}}}}$ is

For a real number $r$,let $[r]$ denote the largest integer less than or equal to $r$. Let $a > 1$ be a real number which is not an integer,and let $k$ be the smallest positive integer such that $[a^k] > [a]^k$. Then,which of the following statements is always true?

The coefficient of $x^{48}$ in $(1+x)+2(1+x)^2+3(1+x)^3+ . . . +100(1+x)^{100}$ is equal to:

The degree of the polynomial $(x+\sqrt{x^4-1})^9+(x-\sqrt{x^4-1})^9$ is

Find $(a+b)^{4}-(a-b)^{4}$. Hence,evaluate $(\sqrt{3}+\sqrt{2})^{4}-(\sqrt{3}-\sqrt{2})^{4}$. (in $\sqrt{6}$)