I'm having a discussion with a co-worker over the number of permutations in a bin packing problems as follows.
There are two bins each of which can hold 6 cu ft. A package can be from 1 – 6 cu feet, there can be from 1 – 12 packages. How many permutations are possible?
It's been a great many years since either of us have done any formal math but it seems to me the problem space isn't all that large due to the constraints, though the problem is NP complete. We found a few web pages talking about different approaches to bin packing but nothing really on how to determine number of possible permutations.
Best Answer
In your problem, each bin is filled independently of the other. Therefore, you need to find out the number of ways you can fill up one bin and square that number.
If $n$ is the capacity of one bin, the number of distinct ways of filling it with packages is given by the partition number $p(n)$. Check out these articles for more details on partition functions
http://en.wikipedia.org/wiki/Partition_%28number_theory%29
http://mathworld.wolfram.com/Partition.html
For your question $n = 6$. Therefore, the number of possible arrangements is $p(n)^2 = 11^2 = 121$.