Question 1184449
The number of ways of lining up 6 objects 3 of which are identical and the other 3 also identical (but different from the previous three) is


{{{6!/(3!3!) = 20}}}.  (You can even list these arrangements!)


Now two such arrangements which satisfy the condition are XOXOXO and OXOXOX.  

The movement of even one letter in either arrangement into another spot will produce 
an arrangement where two adjacent tiles have the same letter on them. This resulting arrangement will belong to the other 18 arrangements.  
In other words XOXOXO and OXOXOX uniquely satisfy the condition of the problem.


Therefore the probability that no two adjacent tiles have the same letter on them is 2/20, or 1/10.