Suppose 7 students are staying in a hall in a hostel and the

Suppose 7 students are staying in a hall in a hostel and they are allotted 7 beds. Among them, Parvin does not want a bed next to Anju because Anju snores. Then, in how many ways can you allot the beds?

Solution

Let the beds be numbered 1 to 7.

This problem is case wise problem and finally we have to add them.

Case 1:

Suppose Anju is allotted bed number 1.

Then, Parvin cannot be allotted bed number 2.

So Parvin can be allotted a bed in 5 ways.

After allotting a bed to Parvin, the remaining 5 students can be allotted beds in 5! ways.

So, in this case the beds can be allotted in 5*5!=600 ways.

Case 2:

Anju is allotted bed number 7.

Then, Parvin cannot be allotted bed number 6

As in Case 1, the beds can be allotted in 600 ways.

Case 3:

Anju is allotted one of the beds numbered 2,3,4,5 or 6

Parvin cannot be allotted the beds on the right hand side and left hand side of Anju