Originally I was just going to ask the problem on my practice math contest, which is asking how many ways there are to write a nine-digit number containing each digit $1-9$ so that the first digit is twice as big as the second and the last two digits are odd. (e.g. $846527913$).
I understood that in this case, the number would be in the form of $EE(EEEOO)OO$ or $EO(EEOOO)OO$ where $E$ is an even, $O$ is an odd, the numbers in the parentheses can be ordered in any way, and the only four cases for the first two are $21, 42, 63,$ and $84.$ Calculating this gave me $240,$ which is definitely wrong (I made a program to check, and it says $7680$).
So here are the questions:
- How do I solve this particular problem correctly? Am I on the right track?
- How are these solved in general? Given that it's a $9$-digit number and each digit $1-9$ is used once, how does one handle constraints such as these? This has always confused me (I'm not very strong in combinatorics)