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Algebra Combinatorics Math Olympiad PRMO

Number of ways of arrangement | PRMO 2017 | Question 10

Try this beautiful problem from the Pre-RMO, 2017 based on Number of ways of arrangement. You may use sequential hints to solve the problem.

Try this beautiful problem from the Pre-RMO, 2017 based on Number of ways of arrangement.

Number of ways of arrangement – PRMO 2017


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.

  • is 107
  • is 48
  • is 840
  • cannot be determined from the given information

Key Concepts


Number of ways

Integers

Arrangement

Check the Answer


But try the problem first…

Answer: is 48.

Source
Suggested Reading

PRMO, 2017, Question 10

Problem Solving Strategies by Arthur Engel

Try with Hints


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.

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