Reasoning behind count of permutations of the word ROOT

Question

If we need to find permutation of words in ROOT it is $4!/2!$ as $2$ objects are the same. Why do we divide by $2!$ and not subtract $2!$?

Answer 1

If we make the $O$'s different, for instance by naming them $O_1$ and $O_2$, then there are $4!$ ways of permuting the word $RO_1O_2T$. Now, these can be paired up, two by two, where each permutation is paired up with the same permutation but with $O_1$ and $O_2$ swapped. Thus $RO_1O_2T$ is paired with $RO_2O_1T$ and $O_1TRO_2$ is paired with $O_2TRO_1$, and so on.

There are $2!$ elements in each pair, so there are $4!/2!$ pairs. Once we make the $O$'s equal again, the two permutations in each pair becomes the same permutation, but permutations from different pairs are still different. Therefore the number of permutations is equal to the number of pairs, and the answer is $4!/2!$.

Subtraction is not used when you "group together" permutations like this, but rather when you have specific permutations that you don't allow. For instance, $4!-2!$ could be the number of permutations of $RO_1O_2T$ that don't spell out the word $ROOT$.