The coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3) \ldots (1+x^{100})$ is

For $2 \le r \le n$,the expression $\binom{n}{r} + 2\binom{n}{r-1} + \binom{n}{r-2}$ is equal to:

$(\sqrt{2} + 1)^6 - (\sqrt{2} - 1)^6 = $

The coefficient of $x^{100}$ in the expansion of $\sum_{j=0}^{200} (1 + x)^j$ is

If the sum of the coefficients in the expansion of $({\alpha ^2}{x^2} - 2\alpha x + 1)^{51}$ vanishes,then the value of $\alpha$ is

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