- Du hast einen Termin bei/mit der Leiterin der Personalabteilung.
Which preposition is appropriate in this sentence?
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Both are valid, but they have slightly different nuances in their meaning:
Termin bei der Leiterin
This includes mit, but usually means that you go to her and meet at her office. Bei most times means, that there is someone else to come to the meeting except when you ask the other person directly like:
Treffen wir uns bei Ihnen?
On the other hand:
Termin mit der Leiterin
You meet her... it is not included where in this statement. This can be your office or her office or anywhere else and if you don't say anything in addition it is normally a personal meeting without anybody else.
Expanding on Torsten's answer, there appears to be a slight semantic difference, too. Having an appointment "bei" someone implies that you come to them and it might also imply that you are not on the same level as them in some way. You don't have appointments "bei" peers, but with people who know more than you or have more power than you. An appointment "mit" implies more equality to me.