but then what happens to the other G?Well, this was how I'd solved it before:
Simply 'choose' 2 Gs out of 3, and the remaining 7 letters can be completely random!
G G _ _ _ _ _ _ _...............[consider GG to be 'attached' together so that it can alternate in 2! ways]
n = 2! x 3c2 x 7!.................. [but don't forget that there are 3 recurring Es too!]
..............3!
...= 5040...... Q.E.D
P.S. If you're still uncertain, let me know. I'll work in it one more time.