Excluding arrangements where letters that are next to each other are identical from total arrangements results in the arrangements where no identical letters are next to one another. There are 3 situations where identical letters are next to one another:
All 3 pairs together
Only 2 pairs together
Only 1 pair together
Having all three pairs next to one another can be expressed as A, A, B, B, C, C (just like in the question). In this case, the number of arrangements is
Having only 2 pairs next to one another can be expressed as C, A, A, B, B, C. The 2 pairs to be together can be selected in , with the third pair being the one where letters are not next to each other. The 2 chosen pairs can swap their places, causing 2 arrangements, each letter from the third pair can be placed in 3 positions, resulting in
Having only 1 pair together (i.e. the other 2 pairs of identical letters will not be next to one another) means that there are to choose a pair of identical letters to be next to one another. On this basis, the number of placements for one pair of identical letters (for example, A) is . When considering all three pairs, this number becomes
The rest involves adding 6, 18 and 36 (resulting in 60) and subtracting this sum from 90, resulting in 30.