Consider equations of the form $x^2 + bx + c = 0$. How many such equations have real roots and have coefficients $b$ and $c$ selected 
from the set of integers $\{1,2,3, 4, 5,6\}$?
$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 19 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 16$
The [discriminant](https://artofproblemsolving.com/wiki/index.php/Discriminant) of the quadratic is $b^2 - 4c$.  Since the quadratic has real roots,
\[b^2 - 4c \ge 0\]
\[b^2 \ge 4c\]
If $b = 6$, then $c$ can be from $1$ to $6$.  If $b = 5$, then $c$ can also be from $1$ to $6$.  If $b=4$, then $c$ can be from $1$ to $4$.  If $b=3$, then $c$ can be $1$ or $2$.  If $b=2$, then $c$ can only be $1$.  If $b = 1$, no values of $c$ in the set would work.
Thus, there are a total of $19$ equations that work.  The answer is $\boxed{\textbf{(B)}}$.