$^{10}C_1 + ^{10}C_3 + ^{10}C_5 + ^{10}C_7 + ^{10}C_9 = $

If $n$ is a positive integer,then $\sum_{r=1}^n r \cdot C_r =$

The sum to $(n + 1)$ terms of the series $\frac{C_0}{2} - \frac{C_1}{3} + \frac{C_2}{4} - \frac{C_3}{5} + \dots$ is

The sum of the series $aC_0 + (a + b)C_1 + (a + 2b)C_2 + \dots + (a + nb)C_n$ is,where $C_r$ denotes the combinatorial coefficient in the expansion of $(1 + x)^n, n \in N$.

Match the expressions in List-$I$ with their values in List-$II$ for the expansion $(1+x+x^2)^n = a_0 + a_1 x + a_2 x^2 + \ldots + a_{2n} x^{2n}$.

List-$I$	List-$II$
$(A)$ $a_0 + a_2 + \ldots + a_{2n}$	$(I)$ $n \cdot 3^{n-1}$
$(B)$ $a_1 + a_3 + \ldots + a_{2n-1}$	$(II)$ $n \cdot 3^n$
$(C)$ $a_1 + 2a_2 + 3a_3 + \ldots + 2n a_{2n}$	$(III)$ $\frac{1}{2}(3^n + 1)$
	$(IV)$ $\frac{1}{2}(3^n - 1)$

The correct match is:

$\binom{47}{4} + \sum_{r=1}^5 \binom{52-r}{3} = \dots$