Traveling Paris to Luserne and back to Paris.

Have you checked Rome2rio for travel times? You’re going to be spending all of your time on the train.
https://www.rome2rio.com/map/Paris/Luzern-station
https://www.rome2rio.com/map/Lucerne-Station/Interlaken
https://www.rome2rio.com/map/Interlaken/Paris

Can you book an open jaw ticket and fly out of Geneva or Zurich?

jeny, you will be spending a whole lot of time just going between all those destinations. Consider a whole day in transit from Paris to Luzern (or wherever). So only the evening there, before you get up and take half a day to get to Interlaken, one evening there etc.

If you have to include Switzerland, I'd spend the whole time in either Luzern or Interlaken, but not both. And not Bern (not in the mountains).

Just pick one base. Luzern has enough to see and do to keep you busy.

And for travel times and prices: use www.sbb.ch/en
(Rome2Rio Is known to be inaccurate at times...)

3 days or 3 nights? What month for this travel?

There are direct trains from Paris to Basel taking 3 hours. You can transfer there to a train to Luzern, adding an hour and a half. That is probably your best choice for a nice Swiss experience in the short time you have.

And I would not go back to Paris on the same day as my flight. It is risky if there is a train delay.

I would pick one. What are you looking to see? mountains and a lake? Interlaken or Lucerne would work for that. I would pick the destination with the fewest connections and shortest time from Paris. I don't suppose you could switch you entry and departure airport to Zurich?

My daughter lives in Luzern. When she goes to Paris, she flies. You might want to check out the time involved and cost of that. Closest airport is Zurich

Flying to Paris from Luzern is not going to save you time. You need to get to Zurich airport first etc... I live in Wengen and when I go to Paris (six times alone last year) I always go by train.

Losts of people default to flights and cars completely unaware that they are not doing themselves a favour that way.

My daughter lives in Luzern. When she goes to Paris, she flies.

Why? I know that North americans tend to see flying as the default mode of transportation, but I really can't see the point in flying from Luzern to Paris when there is a quick and easy train option with one change in Basel.

Tell to take the train next time she goes to Paris.

flight >- 1:15 to Zurich airport; 90 minute checkin; 60 minutes flight; 60 minutes to Paris from whichever airport average (Orly or CDG) = total of 4:45 to 5 hours.

Train, as much as just over 5 hours, as fast as 4:44, and so much easier.