How many arrangements can a team of 8 students have such at all of them have their birthdays in 2 days (but not all on one day) The full question is from:
Find the probability that 8 students in a team will all have their birthday on exactly two days of the week (but not all in one day)?
However, the answers didn't address the main problem, which is finding the total probability. The $\Omega$ is $7^8$ for obvious reasons. So, to find the probability that 8 students in a team have all their birthdays on 2 days (but not all on one day) I thought of this:
Consider: Number of possibilities for 8 students having their birthday all on Sat. or Sun. 
$\Omega($8 students have all their birthdays on say Saturday and Sunday (but not all on one day))

Then first of all we have these possibly combinations of Saturday an Sunday:


*

*7 on Saturday, 1 on Sunday

*6 on Saturday, 2 on Sunday

*...... 1 on Saturday, 7 on Sunday


There are 7 possible "combinations" of how to arrange the players to satisfy the condition that all have their birthdays on Saturday or Sunday (but not all on one day). But of these 7 "combinations" there are 8! permutations we can arrange the students in. So this sample space is $7 \cdot 8!$ (by my reasoning)
So the probability is $\frac{7 \cdot 8!}{7^8}$ 
Now if this reasoning is correct (which I think it is), then I use it to solve the question:
So there are 7 ways to distribute the players to satisfy the condition that they all have their birthday on any two days but not all on one day. For this question it asks for any two days, so the number of ways to choose 2 days from the 7 days of the week is simply: $C_{7,2}$. Of these combinations, there are 7 ways to allocate the players (7 in Day X, 1 in Day Y then 6 in Day X, 2 in Day Y, etc.). So the total number of possibilities is:
$C_{7,2} \cdot 7 \cdot 8!$
However I know that I am wrong since this number is greater than the sample space... but I don't understand what I'm doing wrong, I think my initial reasoning (the "Consider") part is off. Can someone please help me?

I do not think your reasoning is correct. Suppose the two days are fixed on Saturday and Sunday, what we are doing is grouping 8 students into two groups such that no group has zero person. This amount will be $2^8-2$ and not $7\times8!$. So overall the probability should be $\frac{{7\choose2}(2^8-2)}{7^8}$.
In your reasoning, consider the following two permutations:
$1,2,3,4,5,6,7,8$
$8,7,6,5,4,3,2,1$
They will generate duplicated results based on your method of partitioning.(with the first sentence and last sentence re-order each other)
