A person writes letters to six friends and addresses the corresponding envelopes. In how many ways can the letters be placed in the envelopes so that all of them are in a wrong envelope

Explanation

(i) The number of ways in which atleast two of them in the wrong envelopes
`=underset(r=2)overset(6)(sum).^(6)C_(6-r)*D_(r)`
`=.^(6)C_(4)xxD_(2)+.^(6)C_(3)xxD_(3)+.^(6)C_(2)xxD_(4)+.^(6)C_(1)xxD_(5)+.^(6)C_(0)xxD_(6)`
`=15D_(2)+20D_(3)+15D_(4)+6D_(5)+D_(6)` [from note]
`=15xx1+20xx2+15xx9+6xx44+265`
`=719`
(ii) The number of ways in which all letters be placed in wrong envelops`=D_(6)=265`.