Martin spends...........time in museums and libraries looking at old maps and shipping records.

a) a few b) many c) a great deal of d) all

I have chosen b but i'm not sure that right answer is. A great deal of can seem to me the right answer but i don't know exactly

if c is correct why and what is the difference between b and c of course according to the question?
c) is the correct usage. The words "many" and "few" relate to a number, as in "There are many birds in the yard" or " There are a few bananas left." In other words you could substitute a number like "seven" in place of "many" and the sentence would make sense. The phrase "a great deal of . . ." relates to an amount or volume, as in "There is a great deal of snow on the ground." The word "time" cannot not counted, i.e. you can't put a number to it. If, however, the sentence read "Martin spends . . . hours at the museum . . " you would then use "many" or "a few." Hours can be counted.
i see the solution mahlon, i expected so but i wasn't sure now i got it. In that case we can say a lot of instead of a great deal of, can't we?
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That's correct, although I am not a fan of the expression "a lot." There is nothing really wrong with it, but it is rather informal. The meaning, however, would be clear.
Instead of 'a lot of' or 'a great deal of' you could choose 'a good deal of' or 'plenty of' as well.