If $C_j = {}^{n}C_j$,then $C_0 C_r + C_1 C_{r+1} + C_2 C_{r+2} + \ldots + C_{n-r} C_n = $

If ${}^{20}C_{r}$ is the coefficient of $x^{r}$ in the expansion of $(1+x)^{20}$,then the value of $\sum_{r=0}^{20} r^{2} \cdot {}^{20}C_{r}$ is equal to:

If $(1 + x)^n = c_0 + c_1x + c_2x^2 + c_3x^3 + \dots + c_nx^n$,then the value of $c_0 - 3c_1 + 5c_2 - \dots + (-1)^n(2n + 1)c_n$ is

The value of $\sum_{r=0}^{6} \left({}^{6}C_{r} \cdot {}^{6}C_{6-r}\right)$ is equal to:

If ${C_r}$ stands for $^n{C_r}$,the sum of the series $\frac{{2(n/2)!(n/2)!}}{{n!}}[C_0^2 - 2C_1^2 + 3C_2^2 - ..... + {( - 1)^n}(n + 1)C_n^2]$,where $n$ is an even positive integer,is

$\sum_{r=0}^{10} {}^{40-r} C_5$ is equal to