Try this beautiful problem from the Pre-RMO, 2017 based on Number of ways.
Number of ways - PRMO 2017
There are five cities A, B, C, D, E on a certain island. Each city is connected to every other city by road, find numbers of ways can a person starting from city A come back to A after visiting some cities without visiting a city more than once and without taking the same road more than once. (The order in which he visits the cities such as A \(\rightarrow\) B \(\rightarrow\) C \(\rightarrow\) A and A \(\rightarrow\) C \(\rightarrow\) B \(\rightarrow\) A are different).
is 107
is 60
is 840
cannot be determined from the given information
Key Concepts
Number of ways
Integers
Combinatorics
Check the Answer
Answer: is 60.
PRMO, 2017, Question 9
Combinatorics by Brualdi
Try with Hints
A B C D E in this way orderwise such that from A person can visit B,C return to A in $4 \choose 2$ with 2! ways of approach
from A person visits B, C, D comes back to A in $4 \choose 3$ with 3! ways of approach
from A person visits B, C, D, E comes back to A in $4 \choose 4$ with 4! ways of approach
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.
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
Answer: is 48.
PRMO, 2017, Question 10
Problem Solving Strategies by Arthur Engel
Try with Hints
here there is particular way rooms are arranged with guests
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
