If $(1 + x)^n = C_0 + C_1x + C_2x^2 + ... + C_nx^n$,then the value of $C_0 + C_2 + C_4 + C_6 + ...$ is

Let $(1 + x)^m = C_0 + C_1x + C_2x^2 + C_3x^3 + . . . + C_mx^m$,where $C_r = {}^mC_r$ and $A = C_1C_3 + C_2C_4 + C_3C_5 + . . . + C_{m-2}C_m$. Which of the following is false?

If $(1+x-2x^2)^6 = 1+a_1x+a_2x^2+\ldots+a_{12}x^{12}$,then the value of $a_2+a_4+a_6+\ldots+a_{12}$ is

The ratio of the coefficients of the terms $x^{n-r}a^r$ and $x^ra^{n-r}$ in the binomial expansion of $(x+a)^n$ is:

$^{15}C_3 + ^{15}C_5 + \ldots + ^{15}C_{15} = ?$

Let $X = 1({ }^{10} C _1)^2 + 2({ }^{10} C _2)^2 + 3({ }^{10} C _3)^2 + \ldots + 10({ }^{10} C _{10})^2$,where ${ }^{10} C _{ r }$ for $r \in \{1, 2, \ldots, 10\}$ denotes binomial coefficients. Then,the value of $\frac{1}{1430} X$ is: