If the coefficients of $x^4, x^5$ and $x^6$ in the expansion of $(1+x)^n$ are in arithmetic progression,then the maximum value of $n$ is:

$A$ binary sequence is an array of $0$'s and $1$'s. The number of $n$-digit binary sequences which contain an even number of $0$'s is

$\binom{50}{4} + \sum_{i=1}^{6} \binom{56-i}{3} = \dots$

The value of $^{15}C_0^2 - ^{15}C_1^2 + ^{15}C_2^2 - ... - ^{15}C_{15}^2$ is

If $26 \left( \frac{2}{3} \binom{12}{2} + \frac{2}{5} \binom{12}{4} + \frac{2}{7} \binom{12}{6} + \dots + \frac{2}{13} \binom{12}{12} \right) = 3^{13} - \alpha$,then $\alpha$ is equal to:

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: