Let $(1 - 2x + 3x^2)^{10} = a_0 + a_1x + a_2x^2 + \dots + a_n x^n$,where $a_n \neq 0$. Then the arithmetic mean of $a_0, a_1, a_2, \dots, a_n$ is

If $a_k$ is the coefficient of $x^k$ in the expansion of $(1+x+x^2)^n$ for $k=0, 1, 2, \ldots, 2n$,then $a_1+2a_2+3a_3+\ldots+2na_{2n}$ is equal to

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?

If $C_{j}$ stands for ${ }^{n} C_{j}$,then $\frac{C_0}{2} + \frac{C_1}{2 \cdot 2^2} + \frac{C_2}{3 \cdot 2^3} + \ldots + \frac{C_{n}}{(n+1) 2^{n+1}} = $

Find $a, b$ and $n$ in the expansion of $(a+b)^{n}$ if the first three terms of the expansion are $729, 7290$ and $30375$ respectively.

Evaluate the expression: $\frac{1}{81^{n}} - {}^{2n}C_1 \frac{10}{81^{n}} + {}^{2n}C_2 \frac{10^2}{81^{n}} - \dots + \frac{10^{2n}}{81^{n}} = $