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I usually start with 3 then see how my tank is. They need to be in groups. It also depends on what species you get. Bronze corys can get a little big in time.
Depends what kind, most are not really suited for a ten gallon as they are very active and happiest in groups. The dwarf species will work and 5 is a good start.
The dwarf kind to look for are corydoras pygmaeus or corydoras habrosus. There are others but I believe they might be the easier to find. I have heard that panda corys don't get too big but I would want someone to verify that.
I would go with six pygmaeus, they are adorable and much smaller than I expected they would be! I frequently lose them in my tank as their markings make them blend seamlessly into the substrate I have habrusos (never know how to spell that!) and bronze too but I'm not sure I'd put them in a ten gal.