Kim has $10$ identical lamps and $3$ identical tables. How many ways are there for her to put all the lamps on the tables?
Only the number of lamps on each table matters, so we can list the possibilities systematically:  \begin{align*}
(&10,0,0) \\
& (9,1,0) \\
& (8,2,0) \\
& (8,1,1) \\
& (7,3,0) \\
& (7,2,1) \\
& (6,4,0) \\
& (6,3,1) \\
& (6,2,2) \\
& (5,5,0) \\
& (5,4,1) \\
& (5,3,2) \\
& (4,4,2) \\
& (4,3,3)
\end{align*}There are a total of $\boxed{14}$ possibilities.