pLEASE help me with 18th sum
correct option is A,B,C,D

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have} \mathrm{three} \mathrm{elements}: \left\{-2016, 0,2016\right\} \\ \mathrm{There} \mathrm{are} 9 \mathrm{places} \mathrm{to} \mathrm{fill} \mathrm{and} \mathrm{each} \mathrm{place} \mathrm{can} \mathrm{be} \mathrm{filled} \mathrm{with} \mathrm{any} \mathrm{of} \mathrm{the} \mathrm{three} \\ \mathrm{choices} \mathrm{i}.\mathrm{e} \left\{-2016, 0,2016\right\} \\ \mathrm{Hence} \\ \mathrm{Total} \mathrm{matrix} \mathrm{formed}=3^9 \\ \mathrm{Now} \left[\begin{array}{ccc}\&& \&& \&\\ & \&& \&\\ & & \&\end{array}\right] \\ \mathrm{For} \mathrm{symmeric} \mathrm{matrix} \mathrm{we} \mathrm{need} \mathrm{to} \mathrm{care} \mathrm{about} \mathrm{upper} \mathrm{of} \mathrm{lower} \mathrm{triangle} \mathrm{of} \mathrm{the} \mathrm{matrix} \\ \mathrm{as} \mathrm{once} \mathrm{we} \mathrm{fill} \mathrm{upper} \mathrm{triangle}, \mathrm{lower} \mathrm{triangle} \mathrm{becomes} \mathrm{fixed}. \\ \mathrm{So} \mathrm{we} \mathrm{need} \mathrm{to} \mathrm{fill} 6 \mathrm{places} \mathrm{only} \mathrm{nad} \mathrm{remaining} \mathrm{three} \mathrm{becomes} \mathrm{fixed} \\ \Rightarrow \mathrm{Total} \mathrm{number} \mathrm{of} \mathrm{symmetric} \mathrm{matrix}=3^6 \\ \Rightarrow \mathrm{Probabilty} \mathrm{of} \mathrm{getting} \mathrm{symmetric} \mathrm{matrix}=\frac{3^6}{3^9}=\frac{1}{3^3}\mathbf{\to }\mathbf{Option}\left(\mathbf{A}\right) \\ \left[\begin{array}{ccc}0& \&& \&\\ & 0& \&\\ & & 0\end{array}\right] \\ \mathrm{In} \mathrm{skew} \mathrm{symmetric} \mathrm{leading} \mathrm{diagonal} \mathrm{will} \mathrm{be} 0, \mathrm{hence} \mathrm{we} \mathrm{only} \mathrm{need} \mathrm{to} \mathrm{care} \\ \mathrm{about} 3 \mathrm{places} \mathrm{which} \mathrm{can} \mathrm{be} \mathrm{filled} \mathrm{with} \mathrm{any} \mathrm{of} \mathrm{three} \mathrm{choices} \\ \Rightarrow \Rightarrow \mathrm{Total} \mathrm{number} \mathrm{of} \mathrm{skew}-\mathrm{symmetric} \mathrm{matrix}=3^3 \\ \Rightarrow \mathrm{Probabilty} \mathrm{of} \mathrm{getting} \mathrm{skew}-\mathrm{symmetric} \mathrm{matrix}=\frac{3^3}{3^9}=\frac{1}{3^6}\mathbf{\to }\mathbf{Option}\left(\mathbf{B}\right) \\ \mathrm{Trace} \mathrm{is} \mathrm{sum} \mathrm{of} \mathrm{entries} \mathrm{along} \mathrm{leading} \mathrm{diagonal}. \\ \mathrm{For} \mathrm{maximum} \mathrm{value} \mathrm{of} \mathrm{trace}, \mathrm{all} \mathrm{elements} \mathrm{of} \mathrm{leading} \mathrm{diagonal} \mathrm{should} \mathrm{b} \\ \mathrm{filled} \mathrm{with} 2016 \mathrm{and} \mathrm{with} -2016 \mathrm{in} \mathrm{case} \mathrm{of} \mathrm{minimum} \mathrm{trace} \\ {\left[\begin{array}{ccc}2016& \&& \&\\ \&& 2016& \&\\ \&& \&& 2016\end{array}\right]}_{\mathrm{Maximum} \mathrm{trace}} \mathrm{and} {\left[\begin{array}{ccc}-2016& \&& \&\\ \&& -2016& \&\\ \&& \&& -2016\end{array}\right]}_{\mathrm{Minimum} \mathrm{trace}} \\ \mathrm{So} \mathrm{now} \mathrm{we} \mathrm{need} \mathrm{to} \mathrm{fill} 6 \mathrm{remaining} \mathrm{six} \mathrm{places}: \\ \Rightarrow \mathrm{Number} \mathrm{of} \mathrm{matrices} \mathrm{with} \mathrm{maximum} \mathrm{trace}=3^6 \\ \Rightarrow \mathrm{Probabilty} \mathrm{of} \mathrm{getting} \mathrm{matrix} \mathrm{with} \mathrm{maximum} \mathrm{trace}=\frac{3^6}{3^9}=\frac{1}{3^3}\mathbf{\to }\mathbf{Option}\left(\mathbf{C}\right) \\ \Rightarrow \mathrm{Number} \mathrm{of} \mathrm{matrices} \mathrm{with} \mathrm{minimum} \mathrm{trace}=3^6 \\ \Rightarrow \mathrm{Probabilty} \mathrm{of} \mathrm{getting} \mathrm{matrix} \mathrm{with} \mathrm{minimum} \mathrm{trace}=\frac{3^6}{3^9}=\frac{1}{3^3}\mathbf{\to }\mathbf{Option}\left(\mathbf{D}\right) \end{gathered}$$

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