The number of’… is followed by a singular verb.
When we use ‘the number’ we refer to a specific number.
The number of seats reserved is 150.

A number of’… is used with a plural verb.
When we use ‘a number of‘ we mean ‘several’ or ‘more than one’, so a plural verb is used.
Example: A number of passengers were injured.

The number of online customers is increasing.
A number of customers are complaining about slow delivery.
The number of passengers is limited to 50.
A number of people are waiting for the bus.
The number of contaminations is increasing.
A number of those contaminated are in intensive care.
The number of road accidents is causing great concern.
A (large) number of accidents are due to careless driving.
The number of guests at the wedding was quite small.
A number of guests were wearing very stylish clothes.
The number of employees needed is fifteen.
A number of candidates were successful.
The number of those who were successful is 37.
A (surprising) number of candidates have applied.
The number of unemployed people is causing alarm
A number of social problems are caused by unemployment.
The number of people allowed into the museum is limited.
A number of people were refused entry into the museum.
There are a number of new recruits this year.
Here is the number of those who were recruited this year.
There are a number of different options to choose from.
The number of options proposed is considerable.