The coefficient of $x^{100}$ in the expansion of $(1-x)^{-3}$. Please solve it and tell me the technique so that I can solve it in examination in multiple choice questions.

Here is a different approach. We have the geometric series $\frac{1}{1-x}=1+x+x^2+x^3+x^4+...$ Now take the derivative twice and divide left hand right hand side by $2$ to arrive at: $\frac{1}{(1-x)^3}=1+3x+6x^2+...$ which in summation form is $\frac{n^2-n}{2}x^{n-2}$. Since the exponent has to be $100$, it follows that $n=102$ from which the coefficient follows.
