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Find the number of permutations of the word "ATTITUDE" such that no two "T" occur together Using inclusion exclusion principle only.

I found the total number of permutations which is $8!/3!$ then i took both T as a unit "a" thus the number of permutations with 2T's together is 7!.I subtracted them but could not get the right answer.I think that I have overcounted some cases but i dont know which one.Any help would be appreciated.Thanks in advance.

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Unrestricted number of ways to arrange them $ \displaystyle = \frac{8!}{3!}$

Now we need to subtract all the permutations where we have $2$ $T's$ together, that is $7!$ ways

But this also subtracts arrangements where 3rd $T$ comes either before or after $TT$ ($ \, \_TT\_ \,$). So we removed $TTT$ twice instead of removing only once.

So number of desired permutations $\displaystyle = \frac{8!}{3!} - 7! + 6! = 2400$

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