A person writes letters 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 $$x - y =$$

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
$$\eqalign{ & = 6!\left\{ {1 - \frac{1}{{1!}} + \frac{1}{{2!}} - \frac{1}{{3!}} + \frac{1}{{4!}} - \frac{1}{{5!}} + \frac{1}{{6!}}} \right\}\left[ {{\text{Deragement formula}}} \right] \cr & = 360 - 120 + 30 - 6 + 1 = 265 \cr & \therefore x - y = 454 \cr} $$