$\frac{1}{1!(n - 1)!} + \frac{1}{3!(n - 3)!} + \frac{1}{5!(n - 5)!} + \dots = $

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

If $(1 - x + x^2)^n = a_0 + a_1x + a_2x^2 + .... + a_{2n}x^{2n}$,then $a_0 + a_2 + a_4 + .... + a_{2n} = $

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 $P_{n}$ denotes the product of the binomial coefficients in the expansion of $(1+x)^{n}$,then $\frac{P_{n+1}}{P_n}=$

Suppose $\sum_{r=0}^{2023} r \cdot ^{2023}C_r = 2023 \times \alpha \times 2^{2022}$. Then the value of $\alpha$ is $............$