Question 360325
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The other tutor made it unnecessarily complicated.

There are 3 vowels: I, O, E 

and 

there are 4 consonants V, L, N, T

and

there are 4 odd positions: 1st letter, 3rd letter, 5th letter, 7th letter.

So we choose 3 of the 4 odd positions for the vowels in 4C3 or 4 ways.
Having chosen which of the 3 odd positions the 3 vowels are to go in, 
we can then arrange the three vowels in those odd positions in 3P3 
or 3! ways.  
Having placed the vowels, the other 4 positions are filled with the 
4 consonants in 4P4 or 4! ways.

That's 4*3!*4! = 576

Edwin</pre>