Combinatorics are tricky

One of these days I was helping a friend of mine with a relatively simple Combinatorics problem. It goes like this: using the digits from 0 to 9, how many 4-digits natural numbers we can write? There may be repetition of digits, but only if they aren't adjacent: 2424 is a valid number, but 2244 isn't.

The way I solved it was something like this:

• The first digit may be any digit (except 0 or it'll be a 3-digits number actually), so there are 9 possibilities.
• The forth digit may be any digit, so there are 10 possibilities.
• The second number may be any digit, except the first one, so there are 9 possibilities.
• The third number may be any digit, except the second and the fourth, so there are 8 possibilities.

Multiplying it all, we have: 9 × 9 × 8 × 10 = 6480.

According to the book my friend was reading, the answer was 6561, but there wasn't an explanation of why. I suspect the author of the book did something like this:

• The first digit may be any digit, except 0.
• The second digit may be any digit, except the previous.
• The third digit may be any digit, except the previous.
• The fourth digit may be any digit, except the previous.

This way of thinking will yield 9^4 = 6561 and it also seems to be correct. In the doubt of who was right, I did a little Haskell program to check the answer: given the numbers from 1000 to 9999, filter the ones that have adjacent numbers equal and print the length of the list.

module Main
    where

import System.IO

num = [1000..9999]

main = putStrLn $ show $ length $ filter (adjp) num

adjp n = if fd == sd then False
         else
             if sd == td then False
             else 
                 if td == fd then False
                 else True
    where
      fd = (mod n 10)
      sd = (mod (div n 10) 10)
      td = (mod (div n 100) 10)
      ft = (mod (div n 1000) 10)

I don't like the nested ifs and elses, there must be a more elegant way of writing this, but I wrote it quickly and it works, so whatever. If you run it, it will print 6480, meaning that I was correct, but the other way of solving it also seems reasonable, so why it's wrong?