Given that, the front and rear view of a disc.
We have a three-axis, and we can flip with respect to $1-1,2-2$ and $3-3:$
- When we flipped disk with axis ${\color{Blue}{1-1:\text{Front View} \Leftrightarrow \text{Rear View}}}$
- When we flipped disk with axis ${\color{Magenta}{2-2:\text{Front View} \nLeftrightarrow \text{Rear View}}}$
- When we flipped disk with axis ${\color{Purple}{3-3:\text{Front View} \nLeftrightarrow \text{Rear View}}}$
Now we have,
- The number of favorable cases $ = 2$
- The total number of cases $ = 3$
The required probability $ {\color{Green}{= \dfrac{\text{The number of favorable cases }}{\text{The total number of cases}}}} = \dfrac{2}{3}.$
$\therefore$ The probability that the disc ${\color{Red}{\text{DOES NOT}}}$ retain the same front and rear views after the flipping operation is $\dfrac{2}{3}.$
Correct Answer $:\text{C}$