Difference between revisions of "Ball-and-urn"

Revision as of 18:46, 8 February 2016

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 $k$ indistinguishable objects into $n$ bins? We can imagine this as finding the number of ways to drop $k$ balls into $n$ urns, or equivalently to drop $k$ balls amongst $n-1$ dividers. The number of ways to do such is ${n+k-1 \choose k}$ .

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 It is used to solve problems of the form: how many ways can one distribute <math>k</math> indistinguishable objects into <math>n</math> bins? We can imagine this as finding the number of ways to drop <math>k</math> balls into <math>n</math> urns, or equivalently to drop <math>k</math> balls amongst <math>n-1</math> dividers. The number of ways to do such is <math>{n+k-1 \choose k}</math>.
-== Reasoning (One of Several) ==
-If you could only put one ball in each urn, then there would be <math>{n \choose k}</math> 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 <math>n</math> urns (numbered 1 through <math>n</math>) and then you also have <math>k-1</math> urns labeled "repeat 1st", "repeat 2nd", ..., "repeat <math>k-1</math>st". 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 <math>n</math> 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 4 balls and 5 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 2 balls in urn 1, 1 in urn 2, and 1 in urn 4. The same is true for the "repeat" urns options but I use the notation <math>r_1</math> etc.
-*1,2,3,4 <-> 1,2,3,4 (no repeats)
-*1,1,2,4 <-> 1,2,4, <math>r_1</math>
-*1,2,2,2 <-> 1,2, <math>r_2</math>, <math>r_3</math>
-*4,4,5,5 <-> 4,5, <math>r_1</math>, <math>r_2</math>
-Since the reframed version of the problem has n+k-1 urns, and k balls that can each only go in one urn, the number of possible scenarios is simply <math>{n+k-1
-\choose k}</math>
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Difference between revisions of "Ball-and-urn"

Revision as of 18:46, 8 February 2016

Problems