What is the difference between saying "number of times that user accessed the site" vs "total number of times that user accessed the site"?
I feel that they mean the same thing but could not find a resource to support my assumption.
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They very well may.
Context is sort of key here.
If I asked you, "What's the number of times you used the restroom?" there is a nuanced but appreciable difference than, "what's the total number of times you've used the restroom?"
Whereas "what's the number of bugs in the software?" would sound identical to "what's the total number of bugs in the software?" in my opinion, at least.
In your specific example, I'd read it to mean it's asking for the superset. "What's the number of fish at the aquarium?" vs "What's the total number of fish at the aquarium?" The former I'd interpret as the number of species. The latter the count of individuals.