Difference between revisions of "Ball-and-urn"

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The [[ball-and-urn]] technique, also known as '''stars-and-bars''', is a commonly used technique in [[combinatorics]].
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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 <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>.
 
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>.
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*[[1986 AIME Problems/Problem 13]]
 
*[[1986 AIME Problems/Problem 13]]
  
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[[Category:Combinatorics]]
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[[Category:Combinatorics]][[Category:Geometry]]

Revision as of 23:04, 19 June 2013

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 ${n+k-1 \choose k}$.

Problems

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