Difference between A number of and The number of

number of, a—some; several:

• This organization offers a number of books describing rock art sites.

Note: The expression is not equivalent in meaning to the phrase any number of—(also: numbers of ) a sufficiently large number:

• I had any number of reasons not to buy the computer.

number of, the—the quantity or amount of smth.:

• The number of books missing from the library is large.