Hello, pankaj!
There are 8 students from 4 different schools (2 from each school).
In how many ways can these students be put in 4 different rooms (2 students in each room)
so that no room contains students of the same school.
Suppose the students are: .
In how many ways can they partitioned into four unmatched pairs?
This is derangement of four objects.
There are derangements. **
Then the four pairs can be assigned rooms in ways.
Answer: . ways.
** You can count them if you like . . .
@Soroban.Please check my solution.
Number of ways to put the students in rooms such that there are students in every room
Let denote the property that the th room contains students of the same school.
.
.
Therefore,
. Therefore,
.Therefore,
Thus required number of ways=