Simplifying regular expressions
Let's solve this exercise "by induction". Suppose first that the alphabet is only $a,b$. In this case, the only possible words are $ababab\ldots$ and $bababa\ldots$, and you can write a simple regular expressions for them. Now take such a word over the alphabet $a,b,c$. You can break it apart as $w_1 c w_2 c w_3 c \ldots$, in which $w_i \in \{a,b\}^*$. We already have a regular expression describing the $w_i$, so we can write one for the language in question. This approach is general, and you can apply it to any alphabet size.