Solve this following

Let $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots . \mathrm{a}_{10}$ be in G.P. with $\mathrm{a}_{\mathrm{i}}>0$ for $\mathrm{i}=1,2, \ldots ., 10$ and $\mathrm{S}$ be the set of pairs $(\mathrm{r}, \mathrm{k})$, $\mathrm{r} \mathrm{k} \in \mathrm{N}$ (the set of natural numbers) for which

$\left|\begin{array}{lll}\log _{e} a_{1}^{r} a_{2}^{k} & \log _{e} a_{2}^{r} a_{3}^{k} & \log _{e} a_{3}^{r} a_{4}^{k} \\ \log _{e} a_{4}^{r} a_{5}^{k} & \log _{e} a_{5}^{r} a_{6}^{k} & \log _{e} a_{6}^{r} a_{7}^{k} \\ \log _{e} a_{7}^{r} a_{8}^{k} & \log _{e} a_{8}^{\mathrm{r}} a_{9}^{k} & \log _{e} a_{9}^{r} a_{10}^{k}\end{array}\right|=0$

Then the number of elements in $\mathrm{S}$, is :