Let $(1+x+x^2)^{10}=a_0+a_1 x+a_2 x^2+\ldots+a_{20} x^{20}$. If $(a_1+a_3+a_5+\ldots+a_{19})-11 a_2=121 k$,then $k$ is equal to . . . . . . .

Suppose $2-p, p, 2-\alpha, \alpha$ are the coefficients of four consecutive terms in the expansion of $(1+x)^n$. Then the value of $p^2-\alpha^2+6\alpha+2p$ equals

Let $P(x) = 1 + x + x^2 + x^3 + x^4 + x^5$. What is the remainder when $P(x^{12})$ is divided by $P(x)$?

Let $(7 + 4\sqrt{3})^n = p + \beta$,where $n$ and $p$ are positive integers and $\beta \in (0, 1)$. Then $(1 - \beta)(p + \beta)$ is

The coefficient of $x^{10}$ in the expansion of $(1 + x)^2 (1 + x^2)^3 (1 + x^3)^4$ is equal to

Let the coefficient of $x^{r}$ in the expansion of $(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^2+\ldots+(x+2)^{n-1}$ be $\alpha_{r}$. If $\sum_{r=0}^{n-1} \alpha_{r}=\beta^{n}-\gamma^{n}$,where $\beta, \gamma \in N$,then the value of $\beta^2+\gamma^2$ equals: