The coefficient of $t^{20}$ in the expansion of $(1 + t^2)^{10}(1 + t^{10})(1 + t^{20})$ is

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

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.

The coefficient of $t^{24}$ in the expansion of $(1 + t^2)^{12}(1 + t^{12})(1 + t^{24})$ is

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:

Let $\lambda$ be the positive root of the equation $x^2-x-1=0$,and set $a_n = \frac{1}{\sqrt{5}}\left(\lambda^n - (1-\lambda)^n\right)$ for $n \in N$,where $N$ is the set of all natural numbers. Consider the sets $A = \{ n \in N : a_n \text{ is a rational number, but not an integer} \}$ and $B = \{ n \in N : a_n \text{ is an irrational number} \}$. Then: