Cheenta
How Cheenta works to ensure student success?
Explore the Back-Story

# Number of ways | PRMO 2017 | Question 9

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

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

ways=$${4 \choose 2}(2!)+{4 \choose 3}(3!)+{4 \choose 4}(4!)$$

=12+24+24

=12+48

=60.

## Subscribe to Cheenta at Youtube

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

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

ways=$${4 \choose 2}(2!)+{4 \choose 3}(3!)+{4 \choose 4}(4!)$$

=12+24+24

=12+48

=60.

## Subscribe to Cheenta at Youtube

This site uses Akismet to reduce spam. Learn how your comment data is processed.