How many positive integer solutions are there to $x_1+x_2+x_3+x_4<100?$

Question

How many positive integer solutions are there to $x_1+x_2+x_3+x_4<100?$

I know how to approach the problem if it were How many positive integer solutions are there to $x_1+x_2+x_3+x_4=100$, it would be

${n-1 \choose k-1} = {100-1 \choose 4-1} = {99 \choose 3} = 156,849$

But the fact that it's now including all solutions $<100$ is throwing me off. How would I solve this?

Answer 1

Sum the analogous solutions from $4$ to $99$, for instance. You get $$\sum_{k=3}^{98}\binom k3=\binom{99}{4}.$$

Answer 2

For this, you have to introduce a dummy variable say $x_5 > 0$ such that the sum becomes equal to $100$. Then find the number of integral solutions , i.e. when: $x_1 +x_2+ x_3+x_4+x_5=100$ which is nothing but ${99 \choose 4}$