Chapter 3.19, Problem 19.5E

How many five-letter “words” can you make in which no two consecutive letters are the same? A “word” may be any list of the standard 26 letters, so WENJW is a word you would count, but NUTTY is not.

Here is an easy solution: By the list-counting methods of Section 8, the answer is $26\times 25\times 25\times 25\times 25=26\times {25}^4$.

Give a hard solution using inclusion-exclusion, and then show that the two answers are the same.