Is the expression "many fewer combinations" correct? It only gets about 600 hits on Google, against 1,200 for "a lot fewer combinations". What would be a correct way of expressing the idea contained in "many fewer combinations"? would it be best rephrased using something like "a large reduction in the number of combinations", or something similar?
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I would say that the expression "many fewer combinations" is perfectly acceptable and should be understood just fine. As there is some disagreement on this, however, it may be preferable to use a different expression. Some examples of alternates follow.
Regarding the disagreement about many or much fewer, in this discussion, it is contended fewer is an adjective, and thus the adverb much must be used over the determiner many. In my opinion, this is actually a misunderstanding of the function of the word fewer; it is in fact, a determiner itself, not an adjective.
Basically, because "combination" is a countable noun, you use many and fewer to describe it. Uncountable nouns such as "sand" and "water," on the other hand, are described with the words much and less.
However, Google's NGram Viewer shows that while "many fewer" has been in use for quite a while, it did not become the preferred usage until about 1960. This may be due to the reasoning above, and the growth of "many fewer" may be a result of hypercorrection, as Tragicomic suggested in a comment.
Some suggestions that are less awkward about quantifying the quantifier:
the word "fewer" already holds a comparative meaning of being less than "few" considering you mean to say something like "I had few options", "now I have fewer", the word "fewer" alone is enough to signify this. If you want to emphasize a great difference between "few" and "fewer", my suggestion is that you could use "even fewer"