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You can do it, plan train journeys as Europe is well connected by rail network, start from Paris (3 days), proceed to Brussels (1 day), start for Switzerland (3 days) and last to Rome (3 days).
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Europe is an ideal summer vacation spot which needs 10 days to explore a lot. Will share a few Tripoto links written by travellers pertaining to their travel to Europe covering places you have asked for. See if it is suitable for your travel. https://www.tripoto.com/paris/trips/affordable-10-days-trip-to-europe-5d62737568a75 All You Need To Know About Exploring Europe In 10 Days
https://www.tripoto.com/italy/trips/places-to-visit-in-europe Sharing the below Tripoto link just in case it might be helpful for you while travelling. https://www.tripoto.com/europe/trips/planning-to-traveling-in-europe-by-train-some-reasons-why-5f363a982f823 Years back we took a package to four countries and covered Paris, Brussels, cologne and Switzerland for 15 days. There are economical packages available at Thomas cook and SOTC you can check them on their website too.
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I was planning a similar trip last year. The main thing is to choose good hotels along the way. If it is an air travel, the main thing is to book all the airline tickets in advance. And you also need to take care of the transfer from the airport to the hotel. I was lucky, all the countries we planned to visit had offices of the one company https://limousine-service.com/. It's the kind of service I would safely recommend. Be sure to think about excursions to the places you want to see. You can also buy admission tickets to museums, galleries, and attractions in advance. If you do it in advance, it is much cheaper, plus you do not have to stand in line. Good luck.
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