How many combinations are possible that satisfy this criterion? Disclaimer: I'm not a mathematician so I apologise in advance if this comes across poorly written. My suspicion is that this is related to permutations and/or combinations, but it could also be associated with set theory also.
Question: Say a person has 10 footballs (each numbered individually) and 3 bins to store the footballs in (also numbered individually). Each bin must have at least two footballs stored in it, but no more than five footballs stored in it at a given time. How many valid combinations of footballs within bins is possible that satisfy this criterion?
A sample arrangement of the 10 footballs within the 3 bins that satisfies this criterion is as follows:

*

*Bin 1: 5 Footballs (Balls 1, 3, 5, 7 and 9)

*Bin 2: 3  Footballs (Balls 2, 4 and 6)

*Bin 3: 2  Footballs (Balls 8 and 10)

 A: There is no elementary way to do this, but here are two methods.
Exponenetial Generating Functions
There are $3$ labeled bins, $10$ labeled balls, and each bin can hold between $2$ and $5$ balls inclusive. Using the theory of exponential generating functions (see generatingfunctionology, chapter $3$, for information on this method), the number of ways is
$$
10!\times [x^{10}](x^2/2!+x^3/3!+x^4/4!+x^5/5!)^3=37{,}170
$$
Wolfram|Alpha calculation.
Recursive equation
Let $A(n,k)$ be the number of ways to place $n$ balls in $k$ bins, with $2$ to $5$ balls per bin. You can verify that
$$
A(n,k)=\sum_{i=2}^5 \binom{n}{i}A(n-i,k-1)
$$
by considering all of possible placements of balls into the first bin. Combined with the base cases $A(n,1)=1$ when $2\le n\le 5$, and $A(n,1)=0$ when $n\notin \{2,3,4,5\}$, this recursive equation lets you compute the desired value $A(10,3)$ by working up from simpler values. In theory this can be done by hand, but it would be tedious, and can easily be done by a computer program (maybe even Excel). Either way, you get the same answer.
