A number of … are / The number of … is
UsageA number of means several and normally takes a plural verb; the number of refers to one total and takes a singular verb.
A number of … are
Use when
A number of means several or many and is followed by a plural noun and plural verb.
Example
“A number of applications are still under review.”
The number of … is
Use when
The subject is the single total or count, so it takes a singular verb.
Example
“The number of applications is increasing.”
Quick check
a number of cases is, the number of cases are, the number of cases have increased
Usage note