There are $25$ distinguishable students in total. They can be arranged in groups of $5$ such that every group has a unique year of birth shared among all the members of the group (meaning there are $5$ years of birth in total).
The question is: In how many ways can these $25$ students form $5$ groups of $5$ so that there's no group consisting only of students with the same year of birth? The groups aren't labeled in any way.
My solution:
It seems to me that it's a typical Inclusion-Exclusion exercise. Let's denote $S_0$ to be the number of all possible student groups. There are $$S_0 = \binom{25}{5}*\binom{20}{5}*\binom{15}{5}*\binom{10}{5}*\binom{5}{5}$$ such groups, since we can choose $5$ students of $25$ to form the first group, $5$ students of the remaining $20$ to form the second group etc.
Let's now consider the number of groups with one condition applied, that is where exactly one group consists of students all from the same year. Let's denote it by $S_1$ and the number of such groups is $$S_1 = \binom{5}{1}*\binom{20}{5}*\binom{15}{5}*\binom{10}{5}*\binom{5}{5}$$The $\binom{5}{1}$ part is the number of ways in which we can choose a condition. A condition in this case is a single year of birth that all students in a group are going to share. There are $5$ years of birth in total, so we can choose one in $\binom{5}{1}$ ways. By having a full group of which all members have the same year of birth, we now only have to worry about the remaining 20 people, hence the rest of the equation.
Now, applying more conditions is basically the same. $$S_2 = \binom{5}{2}*\binom{15}{5}*\binom{10}{5}*\binom{5}{5}$$ $$S_3 = \binom{5}{3}*\binom{10}{5}*\binom{5}{5}$$ $$S_4 = \binom{5}{4}*\binom{5}{5}$$ $$S_5 = \binom{5}{5}$$
Yielding the final result:
$$R = S_0 - S_1 + S_2 - S_3 + S_4 - S_5$$
Is this a proper application of the Inclusion-Exclusion principle or have I made a mistake somewhere? The exact result doesn't really interest me.
Since the groups are unlabelled, after multiplying the binomial coefficients for each $S_i$ you need to divide by the factorial of however many non-same-birth year groups there are: $$R=S_0/5!-S_1/4!+S_2/3!-S_3/2!+S_4/1!-S_5/0!$$