Find the number of permutations of (1,2,3...n) possessing the following feature:
The number n is the only local maximum e.g. 123465, 123564, 654321,
as opposed to 651432, 261354.
(In reply to re: I guess simplicity is in the mind of the user
by Steve Herman)
Chacun à son gout.
I full-heartedly agree that both approaches are elegant, plausible and simple. So is the puzzle itself.
As to which way is simpler or more straightforward, depends on a personal taste and/or predefined criteria.
If brevity of the proof is one of them please see my "official" definition, where I will try to use minimum of words to expose my (=Jer's) approach.