$100$ can be divided into the sum of several numbers as $1+99$ or $1+1+98$, and we set all the numbers have to be positive. Besides, $1+99$ is different from $99+1$. How many different possibilities are there?
I start to divide $100$ into the sum of two numbers. There are 99 possibilities.
Then divide $100$ into the sum of three numbers. There are $98+97+\dots+1$ possibilities.
Then divide $100$ into the sum of four numbers. For example $1+a+b+c=100$. Then there are $97+96+\dots+1$ ways to write $a,b,c$. For $2+a'+b'+c'=100$, there are $96+95+\dots+1$ ways to write $a',b',c'$.
However I don't think this approach is promising. Is there any feasible way to solve this problem?