Q: In how many ways can the letters in the word STATISTIC be arranged?

Now, I am not interested in the actual answer but rather the method whereby the problem is solved. We can begin by computing the number of arrangements while neglecting the repeated letters, which yields a total number of $\displaystyle 9!$ ways of arrangements.

Now here's is where I am confused: Why do we then followingly divide by the number of arrangements that can be made of the repeated letters? The right answer is obtainable by dividing by the $\displaystyle 2!$ (permutations of S), $\displaystyle 3!$ (permutations of T) and again $\displaystyle 2!$ (permutations of I).

Are we not supposed to take the overcounted arrangements and successively subtract the arrangements of the repeated letters, instead of dividing?