I've been thinking about this problem which I think is interesting, but can't solve it.
There are $n$ distinguishable items, and $b$ distinguishable bins. Each bin has to include at least one item. But, once some set of items are placed in a bin, they become indistinguishable. How many ways are there to place the items into the bins?
(1) The condition that each bin has to include at least one item can be resolved by simply tweaking the problem a bit: suppose there are $n-b$ items, and proceed. So this is not a big hurdle. (Or, it can be, depending on how we handle the second condition below.)
(2) The second condition that the items in a bin are indistinguishable is a bit tricky. Suppose we have 3 items, and 2 bins. The items are numbered as 1, 2 and 3. The bins are denoted as A and B.
The second condition says that, we have to consider the following as identical: A - 1, B - 2, 3. vs. A - 1, B - 3, 2.
However, we have to consider the following as distinct: A - 1, B - 2, 3 vs. A - 2, 3, B - 1.
How can I compute the total number of ways to place the items into the bins?