Hello,
This popped in my head as an interesting problem to know the answer for. I could work this out myself, but I'm hoping someone has it off the top of their head or is more motivated :).
The usual form of the inclusion-exclusion principle or PIE is as follows
Clearly, there are also expressions of the following form
And of this form
Are there closed forms for (bij), (cij) of which especially the latter, I'm sure, could be useful in some problems.
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