Question: How many 4-letter "words" (any combination of 4 letters) have no two consecutive letters identical?
Answer: There are 26 choices for the first letter. Once the first letter is chosen, there are 25 choices for the second letter, then 25 choices for the third letter, then 25 choices for the fourth letter. Hence, there are $26 \cdot 25^3 = 406,250$ such four letter words.
But how can this be? With that, I could make ABBZ for example. But that has a pair of consecutive identical letters. Also, I could make ABBB, which has two pairs of consecutive identical letters. What am I missing? Where does my thinking go wrong?