The mean of the coefficients of $x, x^2, \ldots, x^7$ in the binomial expansion of $(2+x)^9$ 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

If $\sum_{r=0}^{10} \left( \frac{10^{r+1}-1}{10^r} \right) \cdot {}^{11}C_{r+1} = \frac{\alpha^{11}-11^{11}}{10^{10}}$,then $\alpha$ is equal to :

The total number of terms in the expansion of $(x+y)^{100} + (x-y)^{100}$ after simplification is

Find the expansion of $(3 x^{2}-2 a x+3 a^{2})^{3}$ using the binomial theorem.

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