Numbers with no two consecutive equal digits

Text

How many $4$-digit numbers (i.e. from $1000$ to $9999$) have no two consecutive equal digits?

Solution

We count position by position with the product rule.

• First digit: from $1$ to $9$ (it cannot be $0$) $\Rightarrow 9$ choices.
• Each following digit can be any of the $10$ digits except the one immediately before it $\Rightarrow 9$ choices each.

Hence $N = 9\cdot 9\cdot 9\cdot 9 = 9^{4} = \boxed{6561}.$