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 .

Reasoning (One of Several)

If you could only put one ball in each urn, then there would be possibilities; the problem is that you can repeat urns, so this does not work. You can, however, reframe the problem as so: imagine that you have the urns (numbered 1 through ) and then you also have urns labeled "repeat 1st", "repeat 2nd", ..., and "repeat -th". After the balls are in urns you can imagine that any balls in the "repeat" urns are moved on top of the correct balls in the first urns, moving from left to right. There is a one-to-one correspondence between the non-repeating arrangements in these new urns and the repeats-allowed arrangements in the original urns.

For a simple example, consider balls and urns. The one to one correspondence between several of the possibilities and the "repeated urns" version is shown. Since there are 4 balls, these examples will have three possible "repeat" urns. For simplicity, I am listing the numbers of the urns with balls in them, so "1,1,2,4" means balls in urn in urn and in urn The same is true for the "repeat" urns options but I use the notation etc.

• 1,2,3,4 <-> 1,2,3,4 (no repeats).
• 1,1,2,4 <-> 1,2,4, .
• 1,2,2,2 <-> 1,2, , .
• 4,4,5,5 <-> 4,5, , .

Since the re-framed version of the problem has urns, and balls that can each only go in one urn, the number of possible scenarios is simply

Problems

• 2003 AMC 10A Problems/Problem 21
• Mock AIME 3 Pre 2005 Problems/Problem 2
• 2007 AIME I Problems/Problem 10
• 1986 AIME Problems/Problem 13
• 2018 AMC 10A Problems/Problem 11