‘a number of’
A number of things or people means several things or people. You use a plural form of a verb after a number of.
A number of key questions remain unanswered.
An increasing number of women are learning self-defence.
‘the number of’
When you talk about the number of people or things of a particular kind, you are talking about an actual number. After the number of you use a singular form of a verb.
In the last 30 years, the number of electricity consumers has risen by 50 per cent.
When you use number in either of these ways, you can use large or small with it.
His private papers included a large number of unpaid bills.
The problem affects a relatively small number of people.
However, don't use ‘big’ or ‘little’ with number in sentences like these.