"A number of..." is a very common collocation used to denote an indefinite quantity. "The number of..." is much less common, but it could be used to talk about a specific but unstated quantity. He owns a number of houses. (several- an indefinite number) The number of houses that he owns is mind-boggling. (a definite but unstated number)

What is the "DIFFERENCE" between "A number of" and "The number of" A number of = a few or several not a huge/large/big/ amount (in number). 1) A number of arrests were made by police, at the disturbance outside a local McDonalds. 2) The number of people out of work and unemployed, has risen by 15% in this last financial year.

"a number of" is not exact, while "the number of" gives a specific amount. "A number of" means "several" or "a few." and ,"The number of" is about the exact amount.