SOLUTION: please help me with this question, it would be highly appreciated: 3 tiles with the letter X on them and 3 tiles with the letter O on them are placed in a row. The order is cho

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Question 1184449: please help me with this question, it would be highly appreciated:
3 tiles with the letter X on them and 3 tiles with the letter O on them are placed in a row. The order is chosen at random. What is the probability that no two adjacent tiles have the same letter on them?
Thank you taking time out of your day to help me!

Found 2 solutions by ikleyn, robertb:
Answer by ikleyn(52787)   (Show Source): You can put this solution on YOUR website!
.
please help me with this question, it would be highly appreciated:
3 tiles with the letter X on them and 3 tiles with the letter O on them are placed in a row.
The order is chosen at random. What is the probability that no two adjacent tiles have the same letter on them?
Thank you taking time out of your day to help me!
~~~~~~~~~~~~~~~~ 

One such base configuration is  XOXOXO,

with 3! = 6 permutations of X's  and 3! = 6 permutations of O's; in all 6*6 = 36 possible permutations inside this configuration.


Similarly, there are 36 possible permutations inside the other base configuration  OXOXOX.


In all, there are 36 + 36 = 2*36 = 72 favorable permutations among 6! = 6*5^4*3*2*1 = 720 all possible permutations of 6 letters.


In terms of probability, it is  P = 72/720 = 0.1 to have a preferable configuration, randomly.

Solved and explained.



Answer by robertb(5830)   (Show Source): You can put this solution on YOUR website!
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

. (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.
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