1

A Japanese friend wrote to me: なぜsomethingsomethingに興味あるのですか。

  1. Shouldn't she have used が before ある?
  2. Could you just say ありますか instead of あるのですか?
2

「なぜ~に興味あるのですか」 is a perfectly valid sentence, if not very formal.

  1. In formal sentences, yes, が is necessary here. In casual sentences (particularly in speech), particles are often omitted. See: What are the guidelines of omitting particles?
  2. This の is almost mandatory because this sentence is seeking an explanation based on the previous context (which is of course "you are interested in something"). So saying ありますか instead of あるのですか would sound awfully unnatural. Basically の (or ん) is almost always used when you ask a reason using なぜ/どうして.

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