Integer solutions

Question

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

I haven't seen any problems with "less than", so I'm a bit thrown off. I'm not sure if my answer is correct, but if there is, there has a be a more concise form.

Solution:

$$ \sum_{i=0}^{95} \binom{(99-i)+4-1}{99-i} $$

Answer 1

Hint: It is the number of positive solutions of $x_1+x_2+ x_3+x_4 +x_5=100$.

For in how many ways can I distribute candies among $4$ kids, each kid getting one candy at least, and with $\lt 100$ candies distributed?

Imagine I have $100$ candies. I call myself the fifth kid, and if $k$ candies are distributed among the real four, I get the remaining $100-k$. This gives a natural one to one correspondence between distribution of $\lt 100$ candies among $4$ kids, one at least to each, and distributions of $100$ candies among $5$ kids, at least one to each.

Mild modification of the idea takes care of the situation in which we do not have the condition "at least one to each."

Answer 2

Using the hint provided by Andre Nicolas and Brian M. Scott, we add an extra variable $x_5$, then the problem can be viewed as asking for the number of compositions of $n=100$ into $k=5$ parts, or $\binom{100-1}{5-1}$.