Possibly if the kitchen does not have an electric over and electric cook-top and no eclectic water heater in the bathroom. If they are electric have an electrician evaluate what is being powered in the kitchen and the bathroom to determine if a 50 amp sub-panel is large enough. I would for sure at least go with a 60 amp sub-panel.
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A 100-A sub-panel would be fed from a 100-A breaker.
A 200 amp service panel with a 60 amp sub-panel.
Yes. Many installations have breaker totals higher that the main breaker of the panel. It is worked on a percentage basis. Not all of the breakers will be on at the same time. In a home, on a 100 amp panel the average load is 50 - 60 amps. The 100 amp main breaker is protecting the 100 amp rated panel board. If the load current goes higher that the panel board is rated at, the main breaker will trip to protect the board.
You should not load a 125 amp panel any more than 100 amps.
Panels are different. You can get 150 amp panels with 20 to 30 breaker spaces. The limit is stated by the manufacturer on a label attached to the panel. Hopefully you an find the label, sometimes it is on the inside of the cover