$32 \cdot 56$ The three sides of a triangle are positive integers $a, b, c$, and $a \leqslant b \leqslant c$. If it is known that $c=6$, then the number of triangles that can be formed with $a, b, c$ as sides and satisfying the above conditions is
(A) 9.
(B) 10.
(C) 11.
(D) 12.
(China Hubei Province Wuhan City Junior High School Mathematics Competition, 1988)
[Solution] Given $c=6$, and $a \leqslant b \leqslant c$, also $a+b>c$. Then we have the following table
\begin{tabular}{c|llllllllllll}
\hline$a$ & 1 & 2 & 2 & 3 & 3 & 3 & 4 & 4 & 4 & 5 & 5 & 6 \\
\hline$b$ & 6 & 5 & 6 & 4 & 5 & 6 & 4 & 5 & 6 & 5 & 6 & 6 \\
\hline$c$ & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 \\
\hline
\end{tabular}

There are 12 groups in total.
Therefore, the answer is $(D)$.