Let $a_1,a_2,\dots,a_{10}$ be in G.P. with $a_i>0$ for $i=1,2,\dots,10$ and $S$ be the set of pairs $(r,k)$, $r,k\in\mathbb{N}$, for which $ \begin{vmatrix} \log_e(a_1^{\,r}a_2^{\,k}) & \log_e(a_2^{\,r}a_3^{\,k}) & \log_e(a_3^{\,r}a_4^{\,k})\\ \log_e(a_4^{\,r}a_5^{\,k}) & \log_e(a_5^{\,r}a_6^{\,k}) & \log_e(a_6^{\,r}a_7^{\,k})\\ \log_e(a_7^{\,r}a_8^{\,k}) & \log_e(a_8^{\,r}a_9^{\,k}) & \log_e(a_9^{\,r}a_{10}^{\,k}) \end{vmatrix} =0. $ Then the number of elements in $S$, is –