JEE Main · 2004 · 24 Apr 2004 · Offline · Permutations and Combinations

GARDEN has six distinct letters, with the two vowels A and E. All six can be arranged in $6!=720$ ways. In exactly half of these the vowels appear in alphabetical order, A before E, and in the other half they do not, so the count we want is $\frac{6!}{2!}=\frac{720}{2}=360$.