You are right, however... if looked at as a serial outcome, and he only cares about exactly one "five" and the other five results must all be non-fives, then there are only 6 outcomes out of 46656 outcomes: one where the sole five is in the first slot, the second, the third, etc.
There are 46656 total outcomes, WHEN CONSIDERING ALL 6 SIDES of the die. There are only 64 total outcomes when considering only 5s vs. non-5s...which we are here. Of those 64, exactly 6 meet our specification, to wit, exactly one 5, as I illustrated in a previous post.
Don't trust me, but I **should** trust Blaise Pascal, were I you. There is an excellent discussion of (essentially) this very problem in letters between Pascal and the Chevalier du Mere in Peter Bernstein's dandy book, Against The Gods, which is a fine treatise on the history and development of probability theory and risk analysis.