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

If the sum of all the coefficients of $(\alpha x^2 - 2x + 1)^{2019}$ is equal to the sum of all the coefficients of $(x - \alpha y)^{2019}$,then $\alpha = $

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

Which is larger: $(1.01)^{1000000}$ or $10,000$?

If $k$ is a positive integer and $10^k$ is a divisor of the number $9^{11}+11^9$,then the greatest value of $k$ is

If $a$ and $b$ are distinct integers,prove that $a-b$ is a factor of $a^{n}-b^{n}$,whenever $n$ is a positive integer.