Let $A, M,$ and $C$ be [nonnegative integers](https://artofproblemsolving.com/wiki/index.php/Nonnegative_integer) such that $A + M + C=12$. What is the maximum value of $A \cdot M \cdot C + A \cdot M + M \cdot C + A \cdot C$?
$\mathrm{(A) \ 62 } \qquad \mathrm{(B) \ 72 } \qquad \mathrm{(C) \ 92 } \qquad \mathrm{(D) \ 102 } \qquad \mathrm{(E) \ 112 }$
It is not hard to see that 
\[(A+1)(M+1)(C+1)=\]
\[AMC+AM+AC+MC+A+M+C+1\]
Since $A+M+C=12$, we can rewrite this as
\[(A+1)(M+1)(C+1)=\]
\[AMC+AM+AC+MC+13\]
So we wish to maximize
\[(A+1)(M+1)(C+1)-13\]
Which is largest when all the factors are equal (consequence of AM-GM).  Since $A+M+C=12$, we set $A=M=C=4$
Which gives us 
\[(4+1)(4+1)(4+1)-13=112\]
so the answer is $\boxed{\textbf{(E) }112}.$