Question #30daf
1 Answer
The number of distributions of
Explanation:
If empty boxes are allowed, the problem is simple.
Let's model our problem using the following imaginary picture.
Assume, objects are dots on a line (so, we have
Any position of these
This is great, except we did not take into account that objects are identical. That means that any permutation of dots that leaves the mutual configuration of dots and separators unchanged produces the same distribution of objects among boxes. That means, the same distribution we have counted
That results in
Let's analyze it further. The separators also can be permuted without distorting the distribution of objects among boxes. That means that another division must be used to compensate for over-counting. There are
That leaves us with the number of distributions of
This is the final answer.