# 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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#### College Mathematics Program

The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.