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I know that it should be "a number of people are", "the number of people is". We get this:

There are a number of people at the bus stop.

There is the number of people at the bus stop.

What does ARE correspond with? If you say it corresponds with "people" rather than with "a number" then I will ask you why does "is" correspond with "the number"? If it corresponds with "the/a number" it should be "is" in both cases, if it corresponds with "people" it should be "are" in both cases. But it's not the case. Why?

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The suggested sentence

There is the number of people at the bus stop. incorrect, shown by red X

is not correct. If it meant anything at all, it would mean that the number is waiting at the bus stop, not the people, which does not make sense.

The phrase "the number of people " is used only in connection with an actual count, such as:

"the number of people voting 'yes' was twenty-four. correct, shopwn by green check

In this case the referent is to the count, or number, which is singular, not to the people being counted. Thus "is" or "was", not "are" or "were", is used.

"A number of people" is a noun phrase indicating a group of several people. Grammatically, it works in much the same way as "seven people". One would write:

There are seven people at the bus stop. correct, shopwn by green check

and not

There is seven people at the bus stop. incorrect, shown by red X

The agreement is with the noun phrase, which is plural, in either case.

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