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  4. A person writes letlers to six friends and addresses the corresponding envelopes. Let x be the number of ways so that at least two of the letters are in wrong envelopes and y be the number of ways so that all the letters are in wrong envelopes. Then x-y=

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Q. A person writes letlers to six friends and addresses the corresponding envelopes. Let $x$ be the number of ways so that at least two of the letters are in wrong envelopes and $y$ be the number of ways so that all the letters are in wrong envelopes. Then $x-y=$

Permutations and Combinations

Solution:

If all the letters are not in the right envelopes, then at least two letters must be in wrong envelopes.
$\therefore x=6 !-1=719$
Now $y =$ number of ways so that all the letters are in wrong envelopes
$=6 !\left\{1-\frac{1}{1 !}+\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}-\frac{1}{5 !}+\frac{1}{6 !}\right\}$
[Deragement formula]
$=360-120+30-6+1=265$
$ \therefore x-y=454$