The value of the determinant left| begin{arra | vedclass

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(C) Let $\Delta = \left| \begin{array}{ccc} 10! & 11! & 12! \ 11! & 12! & 13! \ 12! & 13! & 14! \end{array} ight|$.Taking $10!$ common from $C_1$,

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$2\,(10!\,11!\,12!)$

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(C) Let $\Delta = \left| \begin{array}{ccc} 10! & 11! & 12! \ 11! & 12! & 13! \ 12! & 13! & 14! \end{array} ight|$.Taking $10!$ common from $C_1$,$11!$ from $C_2$,and $12!$ from $C_3$:$\Delta = 10!\,11!\,12! \left| \begin{array}{ccc} 1 & 11 & 11 	imes 12 \ 1 & 12 & 12 	imes 13 \ 1 & 13 & 13 	imes 14 \end{array} ight|$.Applying $R_2 	o R_2 - R_1$ and $R_3 	o R_3 - R_1$:$\Delta = 10!\,11!\,12! \left| \begin{array}{ccc} 1 & 11 & 132 \ 0 & 1 & 156-132 \ 0 & 2 & 182-132 \end{array} ight| = 10!\,11!\,12! \left| \begin{array}{ccc} 1 & 11 & 132 \ 0 & 1 & 24 \ 0 & 2 & 50 \end{array} ight|$.Expanding along $C_1$:$\Delta = 10!\,11!\,12! 	imes (1 	imes (50 - 48)) = 10!\,11!\,12! 	imes 2 = 2(10!\,11!\,12!)$.

The value of the determinant $\left| \begin{array}{ccc} 10! & 11! & 12! \\ 11! & 12! & 13! \\ 12! & 13! & 14! \end{array} \right|$ is

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