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

In expansion of  $(1+x)(1+x^2)(1+x^3)\mathrm{....}(1+x^{100})$

$x^9$ can be found in the following ways 

$\begin{array}{l}{x}^9,x^{1+8},x^{2+7},x^{3+6},x^{4+5},x^{1+2+6},x^{1+3+5}\\ ,x^{2+3+4}\end{array}$

The coefficient of $x^9$ in each of the above 8 cases is 1.

$\therefore$Required coefficient =8