If the coefficients of $x$ and $x^{2}$ in the expansion of $(1+x)^{p}(1-x)^{q}$,where $p, q \leq 15$,are $-3$ and $-5$ respectively,then the coefficient of $x^{3}$ is equal to $............$

If $C_0, C_1, C_2, \ldots, C_{n}$ are the binomial coefficients in the expansion of $(1+x)^{n}$,then $(C_0+C_1)-(C_2+C_3)+(C_4+C_5)-(C_6+C_7)+\ldots=$

The sum of the coefficients of all even degree terms in $x$ in the expansion of $(x + \sqrt{x^3 - 1})^6 + (x - \sqrt{x^3 - 1})^6$ for $x > 1$ is equal to:

Let $(1 + x + x^2)^{20}(2x + 1) = a_0 + a_1x^1 + a_2x^2 + ... + a_{41}x^{41}$,then $\frac{a_0}{1} + \frac{a_1}{2} + .... + \frac{a_{41}}{42}$ is equal to

Let $C_{r}$ denote the coefficient of $x^{r}$ in the binomial expansion of $(1+x)^{n}$,$n \in N$,$0 \leq r \leq n$. If $P_{n} = C_{0} - C_{1} + \frac{2^{2}}{3}C_{2} - \frac{2^{3}}{4}C_{3} + \dots + \frac{(-2)^{n}}{n+1}C_{n}$,then the value of $\sum_{n=1}^{25} \frac{1}{P_{2n}}$ equals.

Let $M = 2^{30} - 2^{15} + 1$. When $M^2$ is expressed in base $2$,the number of $1$'s in its binary representation is: