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  4. There are 6 letters and 6 directed envelopes. Find the number of ways in which all letters are put in the wrong envelopes.

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Q. There are $6$ letters and $6$ directed envelopes. Find the number of ways in which all letters are put in the wrong envelopes.

Permutations and Combinations

Solution:

$D _{6}=6 !\left[\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}-\frac{1}{5 !}+\frac{1}{6 !}\right]$
$=360-120+30-6+1$
$=391-126=265$