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Coloring problems: ISI MMA 2018 Question 10

This problem is a cute and simple application of the rule of product or multiplication principle in combinatorics,. It appeared in TIFR GS 2018.

Understand the problem

A new flag of ISI club is to be designed with 5 vertical strips using some or all the four colours : green , naroon , red and yellow . In how many ways this can be done  so that no two adjacent strips have the same colour ? 
Source of the problem
Sample Questions ( MMA ) :2019 
Topic
Combinatorics 
Difficulty Level
Medium 
Suggested Book
Schaum’s outline of combinatorics by  Balakrishnan 

Start with hints

Do you really need a hint? Try it first!

This is an application of multiplication property in combinatorics. The rule of product or multiplication principle is a basic counting principle (a.k.a. the fundamental principle of counting). Stated simply, it is the idea that if there are a ways of doing something and b ways of doing another thing, then there are a · b ways of performing both actions.

ISI club has 5 vertical strips . We have to colour them using 4 colours . So , the first strip can be coloured in 4 ways . WLOG we take it to be green . Can the second strip be coloured green ?  No  ! Right ? 
So , we have to choose the second strip from rest of the colours . [ Because two adjacent strip has same colour ] 
Similarly , third strip can be coloured into 3 ways , fourth strips can be coloured into 3 ways and fifth strips can be coloured into 3 ways .[ We have to exclude the colour of the second one]  Therefore , the total number  of probabilities are – \(3^4\) x 4 = 324 

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