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

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

The sum of the coefficients of $x^r$ (where $r=0, 1, 2, \ldots, 15$) in the expansion of $(3x-1)^{15}$ is equal to the sum of the binomial coefficients of which of the following expansions?
$(a)\ (1+x)^{15}$
$(b)\ (1+x)^{16}+(1-x)^{16}$
$(c)\ (1+x)^{16}-(1-x)^{16}$

If $1^2 \cdot \binom{15}{1} + 2^2 \cdot \binom{15}{2} + 3^2 \cdot \binom{15}{3} + \ldots + 15^2 \cdot \binom{15}{15} = 2^m \cdot 3^n \cdot 5^k$,where $m, n, k \in N$,then $m + n + k$ is equal to :-

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