The Number Of...
The rule to remember is this: use a singular verb with the number of ..., but a plural verb with a number of ....
This is because, when we say the number of ..., we are really referring to a number. Since number is a singular head noun, it accordingly takes a singular verb.
But when we say a number of ..., we actually mean some. Hence, a number of becomes a complex determiner. Since the noun that follows of is always plural (e.g. a number of doctors/cases, not a number of *doctor/*case), the verb should also be plural.
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