${{n+2} \choose 15}=209 \cdot {n \choose 13}$

$\begin{array}{l} \left( \begin{array}{l} n + 2\\ \;\,15 \end{array} \right) = 209 \times \left( \begin{array}{l} \,n\\ 13 \end{array} \right)\\ \Leftrightarrow \frac{{(n + 2)!}}{{15!(n + 2 - 15)!}} = 209 \times \frac{{n!}}{{13!(n - 13)!}}\\ \Leftrightarrow \frac{{(n + 2)!}}{{15!(n - 13)!}} = 209 \times \frac{{n!}}{{13!(n - 13)!}}\\ \Leftrightarrow \frac{{(n + 2)!}}{{15!}} = 209 \times \frac{{n!}}{{13!}}\\ \Leftrightarrow \frac{{(n + 2)(n + 1)n!}}{{15 \times 14 \times 13!}} = \frac{{209 \times n!}}{{13!}}\\ \Leftrightarrow \frac{{(n + 2)(n + 1)}}{{15 \times 14 \times 13!}} = \frac{{209}}{{13!}}\\ \Leftrightarrow (n + 2)(n + 1) = 15 \times 14 \times 209\\ \Leftrightarrow (n + 2)(n + 1) = 210 \times 209\\ \Leftrightarrow n = 208 \end{array}$

n+2 choose 15 = 209* n choose 13 gives the equation n^2 + 3n - 43888= 0. Solving for n gives n = 208