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Try this problem from ISI MStat 2018 PSA Problem 14 based on all possible colorings. We provide sequential hints to help you solve the problem.

A flag is to be designed with 5 vertical stripes using some or all of the four colours: green, maroon, red and yellow. In how many ways can this be done so that no two adjacent stripes have the same colour?

- 576
- 120
- 324
- 432

Basic counting principles

But try the problem first...

Answer: is 324

Source

Suggested Reading

ISI MStat 2018 PSA Problem 14

A First Course in Probability by Sheldon Ross

First hint

How many ways you can colour the first stripe and the second stripe?

You have 4 colour options for the first stripe.

You have (4-1) = 3 colour options for the second stripe.

\( 4 \times 3 \).

Second Hint

How many ways you can colour the third stripe and the rest of the stripes?

You have (4-1) = 3 colour options for the second stripe.

You have (4-1) = 3 colour options for the fourth stripe.

You have (4-1) = 3 colour options for the fifth stripe.

Final Step

Therefore, total number of ways to colour = \( (4 \times 3) \times 3 \times 3 \times 3 = 324 ) \)

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