A number of handouts? (8)
I believe the answer is:
thousand
'a number' is the definition.
(I know that thousand is a type of large integer)
'of handouts?' is the wordplay.
'of' indicates an anagram (the answer comes from or 'of' the letters).
'handouts' is an anagram of 'THOUSAND'.
(Other definitions for thousand that I've seen before include "a fair amount?" , "Grand" , "-- Island dressing" , "See 19 Across" , "Cardinal" .)