I'm working on what I think is a very simple problem, but I think I'm approaching it incorrectly.
Background: String length = 8, consists of X, Y, Z (must have one and only one occurrence each) and the digits 0 through 9 (can have any number of occurrences each).
I think I have the right idea for the first part? It asks for the total possibilities. So I made a table with each digit and the possible positions of that digit: \begin{array}{|c|c|} \hline Digit&Possible\ Positions \\ \hline X& 8 \\ \hline Y& 7 \\ \hline Z& 6 \\ \hline Numbers(10)& 5\\ \hline \end{array}
So then the possibilities are $8 * 7 * 6 * 5^{10} = 3,281,250,000$, right? Or do I have the wrong idea?
The second part I found a lot more difficult. It asked how many possibilities if no two letters can be consecutive.
The "X" has 8 possible positions. But after, that, the letters get harder: If my string is X _ _ _ _ _ _ _ , then there are 6 places where Y could go. But, if the string is _ X _ _ _ _ _ _, then there are only 5 places where Y could go.
\begin{array}{|c|c|} \hline Digit&Possible\ Positions \\ \hline X& 8 \\ \hline Y& *** \\ \hline Z& *** \\ \hline Numbers(10)& 5\\ \hline \end{array}
I'm just getting started with this, so simple/intuitive explanations are most welcome! :)