# urns and balls.

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• Jan 19th 2011, 04:16 AM
abhishekkgp
urns and balls.
Let there be n balls and m urns.
1). If all balls are distinct and all urns are also distinct then number of ways in which the balls can be put in the urns is $m^n$

2). If balls are all alike but urns are distinct then the number of ways are $\binom{n+m-1}{n}$

3). If balls are distinct but urns are not then????

4) If both are alike then????

what is the solution to the last two?(Sleepy)
• Jan 19th 2011, 05:12 AM
ahaok
Quote:

Originally Posted by abhishekkgp
Let there be n balls and m urns.
1). If all balls are distinct and all urns are also distinct then number of ways in which the balls can be put in the urns is $m^n$

2). If balls are all alike but urns are distinct then the number of ways are $\binom{n+m-1}{n}$

3). If balls are distinct but urns are not then????

4) If both are alike then????

what is the solution to the last two?(Sleepy)

3). If balls are distinct but urns are not then???? The urns must have a number or so, otherwise you just have One urn!!!
4) If both are alike then???? the same as 3!
• Jan 19th 2011, 05:48 AM
Plato
For #3 you need to know about Stirling numbers of the second kind.

I am not sure exactly what #4 says.
If it means that both balls and urns are indistinguishable, the you need to partition n into m or fewer summons. EXAMPLE: If we have eight identical balls and five identical urns then P(8,5)=18.
• Jan 19th 2011, 07:37 AM
abhishekkgp
Quote:

Originally Posted by Plato
For #3 you need to know about Stirling numbers of the second kind.

I am not sure exactly what #4 says.
If it means that both balls and urns are indistinguishable, the you need to partition n into m or fewer summons. EXAMPLE: If we have eight identical balls and five identical urns then P(8,5)=18.

yes plato. #4 means partitioning.
well, i dont know what are stirling numbers of second kind.
can you suggest me some good text on combinatorics which treats the subject starting form level zero?
i am in college but am not a math major but i want to learn on my own so please help.
• Jan 19th 2011, 07:54 AM
Plato
Click on the link I gave you. That Wikipedia article on Stirling number of the second kind is really quite good.

For #4 and integer partitions, MATHEMATICS OF CHOICE by Ivan Niven has a very readable chapter on that topic.
• Jan 19th 2011, 08:02 AM
abhishekkgp
i want to read combinatorics from the start, like a beginner, because my current understanding is not very good. and i dont know any good text on that? is "MATHEMATICS OF CHOICE" right for me?
thanks for your valuable help.
• Jan 19th 2011, 08:08 AM
Plato
MATHEMATICS OF CHOICE by Ivan Niven was written on what is called Grades 11-14 level in the US. The entire book is very readable. But at the same time it is not trivial.