Try this beautiful problem from the Pre-RMO, 2017 based on Number of ways of arrangement.
There are eight rooms on the first floor of a hotel, with four rooms on each side of the corridor, symmetrically situated (that is each room is exactly opposite to one other room). Four guests have to be accommodated in four of the eight rooms (that is one in each) such that no two guests are in adjacent rooms or in opposite rooms, find number of ways can the guests be accommodated.
Number of ways
Integers
Arrangement
But try the problem first...
Answer: is 48.
PRMO, 2017, Question 10
Problem Solving Strategies by Arthur Engel
First hint
here there is particular way rooms are arranged with guests
Second Hint
Let 1 g be guest in room 1, 3 g be guest in room 3, 6 g be guest in room 6, 8 g be guest in room 8 then arrangement = 1 g 2 empty 3 g 4 empty
5 empty 6 g 7 empty 8 g arrangement wise
where room 1 and room 5 are opposite and facing each other with room 1 has guest and room 5 empty
room 2 and room 6 are opposite and facing each other with room 2 empty and room 6 has guest
room 3 and room 7 are opposite and facing each other with room 3 has guest and room 7 empty
room 4 and room 8 are opposite and facing each other with room 4 empty and room 8 has guest
Final Step
here with four guests to be filled in four rooms
which can be arranged in 4! ways
empty and filled rooms can be arranged in 2! ways
required number of ways=\(2 \times 4!\)=48 ways.
Math Olympiad Program... taught by Olympians and researchers