How many different words can be formed using all the letters of the word GOOGOLPLEX?
Your answer is off by a factor of $2$, seems that you forgot the double letter L.
The word GOOGOLPLEX consists of the letters O (3x), G (2x), L (2x), P (1x), E (1x) and X (1x).
So the number of possible words consisting of these letters is the multinomial coefficient
$$ \binom{10}{3,2,2,1,1,1} = \frac{10!}{3!\cdot 2!\cdot 2!\cdot 1!\cdot 1!\cdot 1!} = 151200. $$
You are almost right.
If $a_1,a_2,\dots$ are the numbers of times different letters appear in your word, then the answer is $n!/a_1!a_2!\dots $. So, in your case the answer is $10!/3!2!2!$ (not including the $1!$'s.