Suman writes letters to his five friends. The number of ways can be letters be placed in the envelopes so that atleast two of them are in the wrong envelopes are

Required number of ways

$=\sum _{\mathrm{r}=2}^5{ }^5{\mathrm{C}}_{5-\mathrm{r}}\mathrm{r}!\left[1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\dots +\frac{(-1{)}^{\mathrm{r}}}{\mathrm{r}!}\right]$

$\begin{array}{c}=\sum _{\mathrm{r}=2}^5 \frac{5!}{(5-\mathrm{r})!}\left[\frac{1}{2!}-\frac{1}{3!}+\dots +\frac{(-1{)}^{\mathrm{r}}}{\mathrm{r}!}\right]\\ =\frac{5!}{3!}\left(\frac{1}{2!}\right)+\frac{5!}{2!}\left(\frac{1}{2!}-\frac{1}{3!}\right)+\frac{5!}{1!}\left(\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}\right)+\frac{5!}{0!}\left(\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}\right)\\ =10+20+(60-20+5)+(60-20+5-1)\\ =10+20+45+44=119.\end{array}$