Ball-and-urn

The ball-and-urn technique, also known as stars-and-bars, is a commonly used technique in combinatorics.

It is used to solve problems of the form: how many ways can one distribute indistinguishable objects into bins? We can imagine this as finding the number of ways to drop balls into urns, or equivalently to drop balls amongst dividers. The number of ways to do such is , given there can be balls in an urn. However, if there can only be a positive amount of balls in an urn, then the number of ways to do such is