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]]. | 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>, given there can be <math>0</math> 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 <math>{n-1 \choose k-1}</math> |
Revision as of 19:31, 27 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 
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 
