There are some grammar tests in a book, Word Power Made Easy, one of which included this question:
'What (kind of, kind of a) car are you looking for?' asked the salesman.
Although I'd have personally preferred "kind of" over "kind of a", I do not see anything wrong in using the latter. Also the answer in the key was the" kind of" version. So, I want to ask why the "kind of" version is certainly correct.
The explanation given in some grammar manuals (like Collins COBUILD English Guides, 1991) is that count nouns can be converted to uncount nouns when they are preceded by expressions like 'a kind of', 'a sort of', even 'a piece of'. These expressions are followed by a noun with no article.
It's very simple. When a customer approaches a salesman, they asks what variety of car are they looking for. So "What kind of a car" literally in sense gives us a meaning, varieties in one car, this is not apt. It should be "what kind of car", which means 'varieties of car available in the showroom'.
