When does one say mit einem and when does one say mit einen?
I am not sure of the difference, either in common usage or grammatically.
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You never say "mit einen", because mit is part of those prepositions that ONLY ONLY ONLY go with Dativ.
Präposition mit Dativ (These are some other preposition that, as well, work only with Dativ)
mit, seit, von, aus, nach, bei, zu
Rememering them by heart will make your life much easier. Whenever you meet them, make the Dativ bell ring :)
"mit einem" :
"mit" is a Dative preposition and "einen" is Accusative, so you can find "mit einen" precisely:
And no for the exception: of course it can technically occur after "mit" but then "mit" is not a preposition but a part of the verb.
Kommst du mit einen Kaffee trinken.
"Einen Kaffee" is connected to "trinken" and has nothing to do with "mit", which is part of "mitkommen".