- Problem Statement
- How many ways are there to arrange 8 different items into 5 different boxes, each containing at least 1 item.

- Relevant Equations
- Number of injections, surjections, and inclusion-exclusion formulas

My teacher solved this using inclusion-exclusion formulas to count the number of surjections from a set of 8 elemets (containing items) to a set of 5 elements (containing boxes). However, I thought of a different solution. But I have a hunch it's wrong.

What I thought is to first make sure every box gets and item, so we are basically counting number of injections from a set of 5 elements(num. of boxes) to the set of 8 elements (number of items). Then we have to arrange the remaining 3 items into 5 boxes. Here we are counting the number of injections from a set of 3 elemets to the set of 5 elemets. In the end, we multiply these two results to obtain the final one.

Please, let me know what you think.

Edit: I have found a mistake in my reasoning. This only makes sure that every box gets at most 2 items. Which doesn't have to be true. One box can contain up to 4 items, respecting the problem setttings.

What I thought is to first make sure every box gets and item, so we are basically counting number of injections from a set of 5 elements(num. of boxes) to the set of 8 elements (number of items). Then we have to arrange the remaining 3 items into 5 boxes. Here we are counting the number of injections from a set of 3 elemets to the set of 5 elemets. In the end, we multiply these two results to obtain the final one.

Please, let me know what you think.

Edit: I have found a mistake in my reasoning. This only makes sure that every box gets at most 2 items. Which doesn't have to be true. One box can contain up to 4 items, respecting the problem setttings.

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