Let a_1, a_2, a_3, …, a_10 be in G.P. with a_i >0 for i =1,2, …., 10 and S be the set of pairs (r, k), r, k ∈ N (the set of natural numbers) for…

Let $\displaystyle \mathrm{a}_1, \mathrm{a}_2, \mathrm{a}_3, \ldots, \mathrm{a}_{10}$ be in G.P. with $\displaystyle \mathrm{a}_{\mathrm{i}}>0$ for $\displaystyle \mathrm{i}=1,2, \ldots ., 10$ and S be the set of pairs (r, k), $\displaystyle r, k \in \mathbf{N}$ (the set of natural numbers) for which $$\left|\begin{array}{llll} \log _e \mathrm{a}_1{ }^{\mathrm{r}} \mathrm{a}_2{ }^{\mathrm{k}} & \log _e \mathrm{a}_2{ }^{\mathrm{r}} \mathrm{a}_3{ }^{\mathrm{k}} & \log _e \mathrm{a}_3{ }^{\mathrm{r}} \mathrm{a}_4{ }^{\mathrm{k}} \\ \log _e \mathrm{a}_4{ }^{\mathrm{r}} \mathrm{a}_5{ }^{\mathrm{k}} & \log _e \mathrm{a}_5{ }^{\mathrm{r}} \mathrm{a}_6{ }^{\mathrm{k}} & \log _e \mathrm{a}_6{ }^{\mathrm{r}} \mathrm{a}_7{ }^{\mathrm{k}} \\ \log _e \mathrm{a}_7{ }^{\mathrm{r}} \mathrm{a}_8{ }^{\mathrm{k}} & \log _e \mathrm{a}_8{ }^{\mathrm{r}} \mathrm{a}_9{ }^{\mathrm{k}} & \log _e \mathrm{a}_9{ }^{\mathrm{r}} \mathrm{a}_{10}{ }^{\mathrm{k}} \end{array}\right|=0 $$ Then the number of elements in S , is :