AMC 10 2017a Problem 10

Joy has $30$ thin rods, one each of every integer length from $1$ cm through $30$ cm. She places the rods with lengths $3$ cm, $7$ cm, and $15$ cm on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How many of the remaining rods can she choose as the fourth rod? $\textbf{(A)}\ 16 \textbf{(B)}\ 17 \textbf{(C)}\ 18 \textbf{(D)}\ 19 \textbf{(E)}\ 20$