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
![](https://espresso.codeforces.com/bce2d5716917a6c2024043b2dda10f53c5fc3bf9.png)
Clearly, there are also expressions of the following form
![](https://espresso.codeforces.com/1183950976ae0f2d421f9b34a96d5e3cd99671ef.png)
And of this form
![](https://espresso.codeforces.com/76c308c9d55c34692bbeb0dfbadcfd3612a22ee0.png)
Are there closed forms for (bij), (cij) of which especially the latter, I'm sure, could be useful in some problems.