If you plan to visit only Le Chateau and its beautiful gardens in Versailles, I would suggest you to go for a day or two, from Paris by suburban train : Versailles in just 13 miles away from Paris.

Normandy is a short train ride : 160 miles from Paris.

I would suggest you to stay in Trouville, which is a lovely place facing Deauville. Frome there, you can easily go to the D-Day beaches or inside the country.