combinations – Preventing a third object of standing next to another 2 in permutations


Find the number of different ways in which the 9 letters of the word GREENGAGE can be arranged if exactly two of the Gs are next to each other.

There two ways to solve this

1)* E * R * E (GG) N * A * E * gives 6 ways for G so 7!/3!×6

2)8!/3! – 2 × 7!/3!

What crossed my mind was answer n2 where I said first I want to know the arrangements where the two Gs will be next to each other 8!/3!

But I was aware that these arrangements may have the third G standing next to the other 2 Gs so I subtracted from them the arrangements where the 3 Gs would be next to each other 7!/3!.

So my answer is 8!/3! – 7!/3! but apparently it’s missing ×2 but why ×2? That’s what I can’t get.

Also, I honestly don’t understand answer n1, as I feel like that when I permutate the 7 letters— while pairing the 2 Gs— without the 3rd G, maybe I will end up with sth like this:

*E * R * E (GG) N * A * E *

(Change the place of the 2Gs)

*(GG) * R * E E N * A * E *

Then replace the first * or second * with a G

G (GG) * R * E E N * A * E *

(The Three Gs next to each other!)