The sum $\sum\limits_{i = 0}^m {\binom{10}{i}} {\binom{20}{m - i}}$,(where $\binom{p}{q} = 0$ if $p < q$),is maximum when $m$ is

If $n$ is an integer greater than $1$,then $a - ^nC_1(a - 1) + ^nC_2(a - 2) + \dots + (-1)^n(a - n) = $

The sum of the series $1 + \frac{1}{2} {}^{n}C_{1} + \frac{1}{3} {}^{n}C_{2} + \dots + \frac{1}{n+1} {}^{n}C_{n}$ is equal to

Choose the correct option regarding the following statements:
$1$. $C_0+C_2+C_4+\ldots+C_n=2^{n-1}$,if $n$ is even
$2$. $C_1+C_3+C_5+\ldots+C_{n-1}=2^{n-1}$,if $n$ is even

If ${C_0}, {C_1}, {C_2}, ......., {C_n}$ are the binomial coefficients,then $2.{C_1} + {2^3}.{C_3} + {2^5}.{C_5} + ....$ equals

If $(1+x)^n = C_0 + C_1 x + C_2 x^2 + \ldots + C_n x^n$,then $C_0 + 2 C_1 + 3 C_2 + \ldots + (n+1) C_n$ is equal to