The value of $\frac{{{C_1}}}{2} + \frac{{{C_3}}}{4} + \frac{{{C_5}}}{6} + .....$ is equal to
$\frac{{{2^n} - 1}}{{n + 1}}$
$n{.2^n}$
$\frac{{{2^n}}}{n}$
$\frac{{{2^n} + 1}}{{n + 1}}$
If $\sum_{ k =1}^{10} K ^{2}\left(10_{ C _{ K }}\right)^{2}=22000 L$, then $L$ is equal to $.....$
${C_0}{C_r} + {C_1}{C_{r + 1}} + {C_2}{C_{r + 2}} + .... + {C_{n - r}}{C_n}$=
If $\mathrm{b}$ is very small as compared to the value of $\mathrm{a}$, so that the cube and other higher powers of $\frac{b}{a}$ can be neglected in the identity $\frac{1}{a-b}+\frac{1}{a-2 b}+\frac{1}{a-3 b} \ldots .+\frac{1}{a-n b}=\alpha n+\beta n^{2}+\gamma n^{3}$, then the value of $\gamma$ is:
Let $(1 + x)^m = C_0 + C_1x + C_2x^2 + C_3x^3 + . . . . . +C_mx^m$, where $C_r ={}^m{C_r}$ and $A = C_1C_3 + C_2C_4+ C_3C_5 + C_4C_6 + . . . . . .. + C_{m-2}C_m$, then which is false
The coefficient of $x^{256}$ in the expansion of $(1-x)^{101}\left(x^{2}+x+1\right)^{100}$ is: