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 . . . . . . .

The coefficient of $x^{37}$ in the expansion of $(1-x)^{30} (1 + x + x^2)^{29}$ is:

Let $\alpha = \sum_{r=0}^{n} (4r^2+2r+1) {}^{n}C_{r}$ and $\beta = \left(\sum_{r=0}^{n} \frac{{}^{n}C_{r}}{r+1}\right) + \frac{1}{n+1}$. If $140 < \frac{2\alpha}{\beta} < 281$,then the value of $n$ is...............

The total number of terms in the expansion of $[(1 + x)^{100} + (1 + x^2)^{100} + (1 + x^3)^{100}]$ is -

Let $R=(5 \sqrt{5}+11)^{2 n+1}$ and $f=R-[R]$,where $[x]$ denotes the greatest integer less than or equal to $x$,then $R f=$

The sum of the series $2 \times 1 \times {}^{20}C_4 - 3 \times 2 \times {}^{20}C_5 + 4 \times 3 \times {}^{20}C_6 - 5 \times 4 \times {}^{20}C_7 + \dots + 18 \times 17 \times {}^{20}C_{20}$ is equal to: