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:

If $(1 + x) (1 + x + x^2) (1 + x + x^2 + x^3) \dots (1 + x + x^2 + \dots + x^n) \equiv a_0 + a_1x + a_2x^2 + a_3x^3 + \dots + a_mx^m$,then $\sum_{r=0}^m a_r$ has the value equal to:

Let $(7 + 4\sqrt{3})^n = p + \beta$,where $n$ and $p$ are positive integers and $\beta \in (0, 1)$. Then $(1 - \beta)(p + \beta)$ is

The coefficient of $x^{50}$ in the expansion of $(1+x)^{100}+2x(1+x)^{99}+3x^2(1+x)^{98}+\dots+101x^{100}$ is:

If $(1 - x + 2x^2)^n = a_0 + a_1x + a_2x^2 + \dots + a_{2n}x^{2n}$,where $n \in N$,$x \in R$,and $a_0, a_1, a_2$ are in Arithmetic Progression $(A.P.)$,then there exists:

If $n$ is an odd positive integer and $(1+x+x^{2}+x^{3})^{n}=\sum_{r=0}^{3n} a_{r} x^{r}$,then $a_{0}-a_{1}+a_{2}-a_{3}+\ldots-a_{3n}$ is equal to