# Fill in numbers on the cube!

You are given a cube. You are told to fill in each vertex with the numbers $$4,5,6,...,11$$, with no repetition. What is the probability that for each two vertices that are connected by a common edge, he two numbers written on them are co-prime?

Source: HK Prelim 2019 Q20

• Note that if it is an ongoing competition you are not supposed to post the problem here. Not before it has ended. – Florian F Jul 3 at 14:33
• @FlorianF It is from 2019. – Culver Kwan Jul 3 at 23:35

That means that up to rotation and reflection, there are only 2 ways to arrange the numbers. The group of symmetries of the cube has size 48, so after rotations and reflections there are $$2\cdot48=96$$ valid arrangements.
There are $$8!=40320$$ ways to arrange the numbers, of which $$96$$ are valid. The probability is therefore $$\frac{96}{40320}=\frac{1}{420}$$.