If the sum of the coefficients in the expansion of $(1 - 3x + 10x^2)^n$ is $a$ and if the sum of the coefficients in the expansion of $(1 + x^2)^n$ is $b$,then:

The number of terms in the expansion of $(x^{2}+y^{2})^{25}-(x^{2}-y^{2})^{25}$ after simplification is

Using the Binomial Theorem,determine which number is larger: $(1.1)^{10000}$ or $1000$.

Let $[x]$ denote the greatest integer less than or equal to $x$. If for $n \in N$,$(1-x+x^3)^n = \sum_{j=0}^{3n} a_j x^j$,then $\sum_{j=0}^{[\frac{3n}{2}]} a_{2j} + 4 \sum_{j=0}^{[\frac{3n-1}{2}]} a_{2j+1}$ is equal to

If the coefficients of $x^2$ and $x^3$ are both zero in the expansion of the expression $(1 + ax + bx^2)(1 - 3x)^{15}$ in powers of $x$,then the ordered pair $(a, b)$ is equal to

If $f(x) = 1 + nx + \frac{n(n-1)}{2} x^{2} + \frac{n(n-1)(n-2)}{6} x^{3} + \ldots + x^{n}$,then $f^{\prime \prime}(1)$ is equal to