If 6 letters are placed at random in 6 addressed envelopes. Then the odds in favour of arranging them such that no letter goes into correct envelope is

$\mathrm{n}\left(\mathrm{S}\right)=6!$

E=no letter go to correct envelope

$\mathrm{n}\left(\mathrm{E}\right)=6! \left(\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}+\frac{1}{6!}\right)$

(De arrangements Property)

$$\begin{gathered} \mathrm{P}\left(\overline{\mathrm{E}}\right)=\frac{360-120+30-6+1}{720}=\frac{265}{720}=\frac{53}{144} \\ \mathrm{P}\left(\overline{\mathrm{E}}\right)=1-\frac{53}{144}=\frac{91}{144} \end{gathered}$$

odds in favour of event $=\mathrm{P}\left(\mathrm{E}\right) : \mathrm{P}\left(\overline{\mathrm{E}}\right)$

                                          $=53 : 91$