A) \[1!\,\,2\,\,1\,\,\,3\]
B) \[1\,\,!\,\,\,3\,\,!\,\,\,5\,\,!\]
C) \[6\,\,!\]
D) \[9\,\,!\]
Correct Answer: C
Solution :
Let \[A=\left| \begin{matrix} 1 & 2 & 3 \\ {{1}^{3}} & {{2}^{3}} & {{3}^{3}} \\ {{1}^{5}} & {{2}^{5}} & {{3}^{5}} \\ \end{matrix} \right|\] \[=1.2.3\left| \begin{matrix} 1 & 1 & 1 \\ {{1}^{2}} & {{2}^{2}} & {{3}^{2}} \\ {{1}^{4}} & {{2}^{4}} & {{3}^{4}} \\ \end{matrix} \right|\] \[=6\left| \begin{matrix} 1 & 1 & 1 \\ 1 & 4 & 9 \\ 1 & 16 & 81 \\ \end{matrix} \right|\] \[=6\left| \begin{matrix} 1 & 0 & 0 \\ 1 & 3 & 5 \\ 1 & 15 & 65 \\ \end{matrix} \right|({{C}_{2}}\to {{C}_{2}}-{{C}_{1}},{{C}_{3}}\to {{C}_{3}}-{{C}_{2}})\] \[=6.3.5\left| \begin{matrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 1 & 5 & 13 \\ \end{matrix} \right|\] \[=90[1(13-5)]\] \[=90\times 8\] \[=720=6!\]
You need to login to perform this action.
You will be redirected in
3 sec
