What does "it's nice to meet your acquaintance" and when shall we use it?

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That should be "It's nice to *make* your acquaintance".  This is very formal but you could say this whenever you meet somebody for the first time.

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It just means that you're happy to meet someone that you have never met before. "Acquaintance" is someone you have met before, so "making your acquaintance" means that you are meeting someone for the first time. It can be said whenever you are meeting someone for the first time.

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"Should" and "shall" are not always interchangeable, but Jerry is right to correct you here. Your question should be "When should we use it?". And the answer, of course, is probably "Never". We say 'Nice to MAKE your acquaintance's, not 'meet'. The phrase 'make someone's acquaintance' means to get to know someone new.

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It is "It is nice to make your acquaintance".

And you can say it whenever you meet someone.

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You should stick with 'should' rather than 'shall'

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