Mathematics - Determinant Question with Solution | TestHub

Let$a_1, a_2, a_3\dots ,a_{10}$be in$G.P.$with$a_i>0$for$i=1, 2, \dots , 10$and$S$be the set of pairs$\left(r, k\right), r, k\in N$(the set of natural numbers) for which

$\left|\begin{array}{ccc}{\log }_e{a}_1^r a_2^k& {\log }_e{a}_2^r{a}_3^k& {\log }_e{a}_3^r{a}_4^k\\ {\log }_e{a}_4^r a_5^k& {\log }_e{a}_5^r{a}_6^k& {\log }_e{a}_6^r{a}_7^k\\ {\log }_e{a}_7^r{a}_8^k& {\log }_e{a}_8^r{a}_9^k& {\log }_e{a}_9^r{a}_{10}^k\end{array}\right|=0$

Then the number of elements in$S,$is: