I’ve been noticing more questions recently that want to “count the number” of something, e.g. count the number of arrangements of a certain type. My initial reaction to this was: “You don’t count the number of arrangements; you count the arrangements.” Then I checked the Ngram Viewer and found that “counts the number of ways” is actually more common than “counts the ways”. The difference is even more pronounced for “count the number of times” vs. “count the times”. Then I checked my own answers on this site and found that I’d written “count the number” so often myself that the search results don’t fit on a page, and that I’d in fact written “count the number of ways” more often than “count the ways”.
This humbling experience made me wonder whether perhaps I was being too prescriptivist and whether “counting the number of” might be a perfectly normal English expression. What do you (especially the native English speakers among you) think – does “count the number of ways” sound right to you? Do you perhaps even prefer it over “count the ways”?