Let $7$ observation be ${x_1},{x_2},{x_3},{x_4},{x_5},{x_6},{x_7}$

$\bar x = 8 \Rightarrow \sum\limits_{i = 1}^7 {{x_i}}  = 56\,\,\,\,\,\,…….\left( 1 \right)$

Also ${\sigma ^2} = 16$

$ \Rightarrow 16 = \frac{1}{7}\left( {\sum\limits_{i = 1}^7 {x_i^2} } \right) – {\left( {\bar x} \right)^2}$

$ \Rightarrow 16 = \frac{1}{7}\left( {\sum\limits_{i = 1}^7 {x_i^2} } \right) – 64$

$ \Rightarrow \left( {\sum\limits_{i = 1}^7 {x_i^2} } \right) = 560\,\,\,\,\,\,\,\,\,…….\left( 2 \right)$

Now, ${x_1} = 2,{x_2} = 4,{x_3} = 10,{x_4} = 12,{x_5} = 14$

$ \Rightarrow {x_6} + {x_7} = 14$ (from $(1)$) and $x_6^2 + x_7^2 = 100$ (from$(2)$)

$\therefore x_6^2 + x_7^2 = {\left( {{x_6} + {x_7}} \right)^2} – 2{x_6}{x_7} \Rightarrow {x_6}{x_7} = 48$