There are 7 letters in the word HEXAGON.

Formula:

Number of permutations of n distinct objects among r different places, where repetition is not allowed, is

P(n,r) = n!/(n-r)!

Therefore, a permutation of 7 different objects in 7 places is

P(7,7) = = = = 5040.

They can be permuted in P (7,7) = 5040 ways.

The vowels in the word are E, A, O.

Consider this as a single group.

Now considering vowels as a single group, there are total 5 groups (4 letters and 1 vowel group) can be permuted in P (5,5)

Now vowel can be arranged in 3! ways.

Formula:

Number of permutations of n distinct objects among r different places, where repetition is not allowed, is

P(n,r) = n!/(n-r)!

Therefore, the arrangement of 5 groups and vowel group is

P(5,5)×3! = = × 3! = × 6 = 720.

Hence total number of arrangements possible is 720.