Thank you for trying, but you're answer is not correct. The correct answer was 0.913.
We can use complementary counting to solve this problem, by subtracting the probabilty that no adjacent faces are the same color:
1: All faces are colored diffferently. 6!=720 ways
2: One pair of opposite faces are colored the same. 3 ways to choose the pair of faces * 6 ways to color the faces * 120 ways to color the remaining faces = 2160 ways.
3: Two pairs of opposite faces are colored the same. 3 ways to choose the pairs of faces * 30 ways to color the faces * 12 ways to color the remaining faces = 1080 ways.
4: Each pair of opposite faces are the same. 6 * 5 * 4 ways to choose the colors = 120 ways.
(720 + 2160 + 1080 + 120) / 6^6 (total number of ways) ~ 0.08744.
1 - 0.08744 ~ 0.913.
Thank you for trying, and I hope this helped!
