The number of vs A number of

[The Exalted Father]

(a) -The number of students in the class is fifteen.

(b) - A number of students are late.

Could you please differentiate between the usage of verb Be in “a” and “b”?

[5jj]

The number of refers to a figure. That figure is singular, even though it often has a plural reference. A number of, despite its singular appearance, has a similar meaning to some/several, and is followed by a plural noun, which functions as the subject of the verb