<?
/*
You have 5 balls that you have to divide (put the balls in the cups) between a number of cups in all possible ways.

1 cup:
---
|5|
---
1 possibility
Total: 1 possibility

2 cups:
-----
|1|4|
-----
|2|3|
-----
|3|2|
-----
|4|1|
-----
|5|0|
-----
|0|5|
-----
6 possibilities
Total: 7 possibilities

3 cups:
21 possibilities
Total: 28 possibilities

Continue until you have 500 cups. The total of possibilities from 1 to 500 is the answer.
*/
for($cups=1; $cups <= 500; $cups++) {

}
unfinished
?>