[Solution] From the problem, we note that for the equation to have real roots, it must be that $\Delta=b^{2}-4 c \geqslant 0$. After calculation, we obtain the following table:
\begin{tabular}{c|c|c|c|c|c|c}
\hline Values of $c$ & 6 & 5 & 4 & 3 & 2 & 1 \\
\hline Values $b$ can take & 5,6 & 5,6 & $4,5,6$ & $4,5,6$ & $3,4,5,6$ & $2,3,4,5,6$ \\
\hline
\end{tabular}

In total, there are 19 possibilities.
Therefore, the answer is $(B)$.
$2 \cdot 29$ If the coefficients $b$ and $c$ of the equation $x^{2}+b x+c=0$ are elements of the set $\{1,2,3,4,5, 6\}$, then the number of equations that have real roots is
(A) 20.
(B) 19.
(C) 18.
(D) 17.
(E) 16.
(14th American High School Mathematics Examination, 1963)