The value of `|1 1 1^n C_1^(n+2)C_1^(n+4)C_1^n C_2^(n+2)C_2^(n+4)C_2|` is

Explanation

$$\left|\begin{array}{ccc}1 & 1 & 1 \\ { }^{n} C_{1} & { }^{n+2} C_{1} & { }^{n+4} C_{1} \\ { }^{n} C_{2} & { }^{n+2} C_{2} & { }^{n+4} C_{2}\end{array}\right|$$
$$\Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ n & n+2 & n+4 \\ n^{2}-n & n^{2}+3 n+2 & n^{2}+7 n+12\end{array}\right|$$
$$c_{1} \rightarrow c_{1}-c_{2}$$
$$c_{2} \rightarrow c_{2}-c_{3}$$
$$\Delta=\left|\begin{array}{ccc}0 & 0 & 1 \\ -2 & -2 & n+4 \\ -4 n-2 & -4 n-10 & n^{2}+7 n+12\end{array}\right|$$
$$\Delta=1 / 2[8 n+20 \cdot 8 n-4]$$
$$\Delta=8$$