With two slits experiment shooting electrons when we place a detector on one of the slits the electrons act as particles.
What will happen if we use three splits and one detector? Will we have an interference pattern? If we do how will it look like?
Also in general if we have $n$ slits and $m; m < n - 1$ detectors how would the pater look like?
If there is a detector at one of the slits, it corresponds to a measurement with the associated eigenvalues, say, $1$ (the particle passes through the given slit) and $0$ (the particle doesn't pass through the given slit). Now if there are two other slits, the eigensubspace corresponding to the eigenvalue $0$ would be two-fold degenerate corresponding to the states of particle passing through slit $2$, slit $3$, and superpositions thereof.
Thus, if the detector measures $1$ then the particle would not be in any of the "slit-superpositions" and wouldn't interfer with itself. It simply go to the screen and produce a diffraction pattern. But if the detector measures $0$ then the degeneracy of the eigensubspace will make sure that the particle would still be in a superposition of going through slit $2$ and slit $3$ and thus, it would interfere with itself and produce a double-slit interference pattern on the screen.
So when you do the experiment, the end result would be that the pattern on the screen would be a simple summation of the double-slit interference pattern corresponding to the slits without detectors and the diffraction pattern corresponding to the slit with a detector.
I'll leave the generalization of this to you which should is easy enough. :)
With two slits experiment shooting electrons when we place a detector on one of the slits the electrons act as particles.
That isn’t what happens. Instead of slits the first experiments were done with one wire only. No slits at all. The electrons get bended around it and on the screen they saw an intensity distribution:
What will happen if we use three splits and one detector? Will we have an interference pattern? If we do how will it look like?
Single edges are enough to get an intensity distribution with electron-poor and with electron-rich areas. Some years earlier single edge deflection was observed.
Fresnel electron diffraction structures at an edge
Also in general if we have $n$ slits and $m; m < n - 1$ detectors how would the pater look like?
Take it this way: Whenever you prevent - for example by a measurement or the close of a slit - the electrons to interact with edges, the intensity distribution pattern disappear for that edge / edges / slit. The other edges or slits work as awaited, they deflect the electrons and the superposition of all edges deflections are on the screen.
