SOLUTION: Five paychecks and envelopes are addressed to five different people. The paychecks are randomly inserted into the envelopes. What is the probability that at least one paycheck is i

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Question 435922: Five paychecks and envelopes are addressed to five different people. The paychecks are randomly inserted into the envelopes. What is the probability that at least one paycheck is inserted in the correct envelope?
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
Five paychecks and envelopes are addressed to five different people. The paychecks are randomly inserted into the envelopes. What is the probability that at least one paycheck is inserted in the correct envelope?


The answer is 1 minus the probability that all checks go wrong.

The number of ways n things in n original positions can be 
"deranged", that is, arranged so that none of the n things 
are in their original positions is a number called 
"n-subfactorial". This is written !n, and the formula for 
it is

 
   
Therefore the number of ways all 5 checks go wrong is



So the numerator for the probability that all the 
checks go wrong is 44.

The denominator is the number of different ways the 
checks can be placed in any envelopes is 5! or 120.

Therefore the probability that all checks go wrong 
is 44/120 = 11/30.

Therefore the probability that at least one went 
right is

1 - 11/30 = 30/30 - 11/30 = 19/30.

Edwin
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