How many functions $f:\{a,b,c,d,e\} \rightarrow \{1,2,3,4,5\}$ satisfy the following two properties:
$f(a)+f(b)+f(c) = 8$
$f(d)+f(e) = 6$
Using the stars-and-bars method of sorting, I was able to identify that there are 8 "ones" that need to be sorted in the first equation, and 6 in the second. The first has 3 categories (a,b,c) and the second has 2.
Because $f(a)$, $f(b)$, $f(c)$ must each be at least $1$, we know that we have 5 remaining "ones" to sort out. We can solve this problem (5 remaining ones, 2 different categories, 3 already distributed ones) using $\binom{7}{2}$.
We can apply this same strategy for the second property, to get $\binom{5}{1}$. If we multiply these, we get $105$ different functions that satisfy these properties.
I am concerned that some of these $105$ are not actually viable since there must be some combination of function that overlaps the available numbers, i.e., some kind of intersection that needs to be subtracted? How would one go about doing this?
The set $ \{a,b,c \}$ can be mapped onto the sets $\{1,2,5\},\{1,3,4\},\{2,2,4\},\{2,3,3\}$ in respectively $6,6,3,3$ ways. ($18$ in total).
The set $\{d,e\}$ can be mapped onto the sets $\{1,5\},\{2,4\},\{3,3\}$ in respectively $2,2,1$ ways. ($5$ in total).
So there are $18 \times 5= \color{red}{90}$ possible functions.