Update
I was reading the comments to the question, and found that @Kuba had already provided the following answer. I think it's the cleanest solution, so it deserves to be an answer, but please credit him with the idea.
Another idea is to use PatternSequence:
ClearAll[f]
f[PatternSequence[a_,b_,c_]?NumericQ] := a+b+c
Examples:
f[1,2,3]
f["a",2,3]
6
f["a", 2, 3]