# Fill in numbers on the cube … again!

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

Source: a mistaken understanding of that other question: Fill in numbers on the cube!

• Wouldn't the answer depend on what numbers you choose to write? – msh210 Jul 6 at 21:40
• They are chosen randomly. – Florian F Jul 7 at 6:59
• "You are told to fill in each face with some of the numbers" does not sound like they're chosen randomly! I really recommend you edit the question to clarify. – msh210 Jul 7 at 13:21

Rotate the cube so that the 4 is on top, and the 5 in front. The bottom face must be the 8, and the other three odd numbers are coprime so can go in any order on the remaining faces. Up to rotation this gives $$3!=6$$ essentially different arrangements. With rotations this gives $$6\cdot24=144$$ valid number arrangements.
We need to find the total number of possible arrangements. First choose 6 of the 8 numbers, and then arrange them on the faces in any order. That gives $$\binom{8}{6}\cdot6!=20160$$ ways.
The probability then becomes $$\frac{144}{20160}=\frac{1}{140}$$.