Question: In calculus and differential equations, a standard example of word problems are mixing problems, with some number of tanks, and brine often being an output of the system. With one tank, I can imagine some relation to real world scenarios, as people actually make brine, or maintaining aquariums (perhaps not varying salt content, but doing something like controlling pH).

What I have more trouble motivating with good concrete practical applications is systems where multiple tanks are involved and the tanks flow into each other at possibly different rates, though no doubt there are many examples from environmental sciences, chemistry, etc. Can someone provide convincingly practical mixing examples involving multiple tanks (loosely interpreted)?

Answer: 
Your indicator function is just a Kronecker delta $\delta\_{m+n,i+j}$. The Kronecker delta admits an integral representation
$$
\delta\_{a,b}=\int\_{0}^{2\pi}\frac{d\xi}{2\pi}e^{i\xi (a-b)}\ ,
$$
therefore your sum is
$$
\sum\_{m=0}^{N} \sum\_{n=0}^N \sum\_{i=0}^N \sum\_{j=0}^N \mathbb{1}(m,n,i,j)=
\int\_{0}^{2\pi}\frac{d\xi}{2\pi}\sum\_{m=0}^{N} \sum\_{n=0}^N \sum\_{i=0}^N \sum\_{j=0}^N e^{i \xi (m+n-i-j)}
$$
$$
=\int\_{0}^{2\pi}\frac{d\xi}{2\pi}\left(\sum\_{m=0}^{N} e^{i\xi m}\right)^2 \left(\sum\_{j=0}^{N} e^{-i\xi j}\right)^2 =\int\_{0}^{2\pi}\frac{d\xi}{2\pi}\left(\frac{-1+e^{i N \xi +i \xi }}{-1+e^{i \xi }}\right)^2\left(\frac{e^{-i N \xi } \left(-1+e^{i N \xi +i \xi }\right)}{-1+e^{i \xi }}\right)^2=\frac{1}{3} \left(2 (N+1)^3+N+1\right)
$$
Given any $m,i$ there are $N+1-|m-i|$ pairs $n,j$ such that $m+n=i+j$ and so
\begin{eqnarray}
\sum\_{m=0}^N \sum\_{i=0}^N \sum\_{j=0}^N \sum\_{n=0}^N 1(m,n,i,j) &=& \sum\_{m=0}^N \sum\_{i=0}^N ( N+1-|m-i| ) \\
&=& (N+1)^3 - \sum\_{m=0}^N \sum\_{i=0}^N |m-i| \\
&=& (N+1)^3 - 2\sum\_{m=0}^N \sum\_{i=0}^m (m-i) \\
&=& (N+1)^3 - 2\left( \sum\_{m=0}^N \sum\_{i=0}^m m - \sum\_{m=0}^N \sum\_{i=0}^m i \right)\\
&=& (N+1)^3 - 2\left( \sum\_{m=0}^N m (m+1) - \sum\_{m=0}^N {1 \over 2} m (m+1) \right)\\
&=& (N+1)^3 - \sum\_{m=0}^N m (m+1)\\
&=& (N+1)^3 - {1 \over 3} N (N+1)(N+2)\\
&=& {1 \over 3} (N+1)(2N^2+4N +3)
\end{eqnarray}