Ram painted the six faces of a cube with 6 different colors- Red, yellow, white, blue, green and black. What is the probability that the faces with the colors-Red ,yellow and white have a corner in common?
My take:
Total number of ways of painting the cube with these $6$ colors is $\frac{720}{8} =90$. How do i proceed from here? Could anyone please help. Thank you
Best Answer
The total number of distinguishable possible cubes is $30$.
This is because if you fix one colour, you can choose $5$ to go on the opposite face, and the remaining four can be arranged in a circle in $\frac{4!}{4}=6$ ways, allowing for rotations of the same configuration.
As for the numerator of this probability, you can arrange the three chosen colours in a circle in $\frac{3!}{3}=2$ ways, and independently arrange the remaining three colours in $3!=6$ ways, so the total is $12$ desired configurations.
So the answer is $\frac{12}{30}=\frac25$