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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
