I have a question about an 8 sided die problem. I will put up my work what I have if someone can tell me how to proceed I will appreciate it.
We roll an 8 sided die numbered 1 to 8 six times and record the result of the face value that pops up on top as $x_{1}, x_{2} , x_{3}, x_{4}, x_{5}, x_{6}$.
We must now find the total number of ways that $x_{1} \ge x_{2} \cdots x_{5} \ge x_{6}$.
My progress:
If we had $6$ numbers then there will be exactly one way to arrange it such that they are going from increasing to decreasing.
Like what I mean is if the numbers that showed up were 1 through 6 then there are 720 orientations of these numbers but only one orientation will put them in increasing order.
So I thought it would be $8^6/6!$ since for every 720 combinations one puts them in increasing order, but that does not even turn out to be an integer so I am not sure what I am doing wrong.
Thanks in advance!