SOLUTION: You own 12 CDs. You want to randomly arrange 5 of them in a CD rack. What is the probability that the rack ends up in alphabetical order?

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Question 1204448: You own 12 CDs. You want to randomly arrange 5 of them in a CD rack. What is the probability that the rack ends up in alphabetical order?
Answer by ikleyn(52809) About Me  (Show Source):
You can put this solution on YOUR website!
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You own 12 CDs. You want to randomly arrange 5 of them in a CD rack.
What is the probability that the rack ends up in alphabetical order?
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The number of all different ordered quintuples of five CDs from twelve CDs is  

    total = 12*11*10*9*8.    (1)


The number of all different quintuples of 5 CDs from 12 CDs is 

    C%5B12%5D%5E5 = 12%21%2F%285%21%2A7%21%29 = %2812%2A11%2A10%2A9%2A8%29%2F%281%2A2%2A3%2A4%2A5%29  

(without looking in ordering - same as the number of combinations  C%5B12%5D%5E5).



The number of all different ORDERED quintuples of 5 CDs from 12 CDs is 

    5%21%2AC%5B12%5D%5E5 = 5%21%2A%2812%21%2F%285%21%2A7%21%29%29    (2)

(without looking in ordering).  Notice that number (2) is the same as number (1).



When we estimate the number of alphabetically ordered quintuples among all ordered quintuples (2),
we should divide the number (2) by 5!, because among 5! permutations only one is ordered alphabetically.


So, from the number of all ordered quintuples (1) or (2), the number of those ordered alphabetically is

    favorable = 12%21%2F%285%21%2A7%21%29 = %2812%2A11%2A10%2A9%2A8%29%2F%281%2A2%2A3%2A4%2A5%29.    (3)


The problem asks about the probability  

    P = favorable%2Ftotal = alphabetically_ordered_quintuples%2Fall_different_ordered_quituples =  = 1%2F%281%2A2%2A3%2A4%2A5%29 = 1%2F5%21 = 1%2F120.    ANSWER

Solved.