$n$ men are getting on a plane which contains $n+k$ seats. Each one has a seat number but among them, $m$ men forgot his seat number. They get on the plane one by one. For person $X$ if he knows his seat number and if the seat is empty then he take it but if the seat if occupied then he chooses randomly a chair and sits. On the other hand if $X$ does not know his seat number them he chooses randomly a chair and sits. What is the probability that the last $i$ persons sit on their proper seat!? (Those who forgot their seat number are not necessary the first $m$ persons who get on the plane)
COMMENT: Some versions of this problem are posted to SE. You can see the following ones: Number 1, Number 2, Number 3