Unique ways to distribute k1, k2, .. colored balls into n boxes uniquely

Asked Jul 02 '15 at 07:06

Active Jul 02 '15 at 11:40

Viewed 379 times

Example: Uniquely distribute 2 Red Balls and 4 Blue Balls into 3 boxes:

[B][BB][RRB]
[B][BBB][RR]
[B][R][RBBB]
[B][RB][RBB]
[BB][R][RBB]
[BBB][R][RB]

Answer:

6 combinations

[B][B][RRBB] is not valid. Each box should be unique from other boxes. [RRB][BB][B] is not valid. This is same as the first one.

So, how to arrive at (an algorithm is also good) unique ways to distribute N (= k1 + k2 + k3 .. km) balls into n boxes so that no two boxes has same combination. where, ki balls are indistiguishable from each other, but distiguitable from other N-ki balls. (i = 1..m)

Went through below links with not much help:

http://scollyer-tuition.blogspot.in/2011/03/permutations-and-combinations-using.html

https://en.wikipedia.org/wiki/Hypergeometric_distribution#Multivariate_hypergeometric_distribution

Unique ways to keep N balls into K Boxes?

edited Apr 13 '17 at 12:20

Community

asked Jul 02 '15 at 07:06

Sanjay Vamja

1

Something is wrong with the first row in the grey box. – zoli Jul 02 '15 at 07:13
Why are you citing [RB][BB][B] (which has only 5 balls) in the text as invalid and "same as the first one" ? – true blue anil Jul 02 '15 at 07:51
right, @trueblueanil. my bad. correcting it, right away! – Sanjay Vamja Jul 02 '15 at 08:34
Is [ RBBB ] [ RB ] [ ] valid? – Henry Jul 02 '15 at 14:16
This is the same as the number of ways to factor $2^{k1}3^{k2}5^{k3}...$ into $n$ distinct factors. – Empy2 Jul 02 '15 at 16:07
That's true @michael. I traced came from there. But how to get to the solution? – Sanjay Vamja Jul 08 '15 at 11:44
@Henry, Actually I want a solution including blank box. But then I realized that [n boxes with blank box] = [n-1 boxes without blank box] + [n boxes without blank box]. – Sanjay Vamja Jul 08 '15 at 11:46

Unique ways to distribute k1, k2, .. colored balls into n boxes uniquely

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