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A large number of passengers is waiting for the bus at the bus stop.

Is this correct sentence grammatically? Is "passengers" correct in the context as they are not traveling so far?


No one is there at the bus stop.

Is this correct?

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A large number of passengers is are waiting for the bus at the bus stop.


Not quite: "passengers" is plural, so the verb should be the plural "are".

Note that "number" when used with the indefinite article "a" always takes a plural noun as object of the preposition "of", and thus always takes plural verbs.

Your second example is OK.