3 colors of paint!

Probability Level 2

How many different ways are there to color a 3 × 3 3\times3 grid with green, red, and blue paints, using each color 3 times?

Try other painting n × n n\times n grid problems .


The answer is 1680.

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Geoff Pilling by Geoff Pilling , 53, USA

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5 solutions

Andy Hayes
Jun 1, 2016

Relevant wiki: Permutations with Repetition

There are 9 possible squares to paint. There are 3 distinct colors that will be painted on 3 squares each. This is a case of Permutations with Repetition.

The number of permutations is 9 ! 3 ! 3 ! 3 ! = 1680 \dfrac{9!}{3!3!3!}=\boxed{1680} .

13 Helpful 2 Interesting 1 Brilliant 4 Confused
Geoff Pilling
Apr 22, 2016

There are 9 ! / ( 3 ! ∗ 3 ! ∗ 3 ! ) = 1680 9!/(3!*3!*3!) =1680 ways.

7 Helpful 0 Interesting 1 Brilliant 1 Confused

comb(9,3) * comb(6,3) * comb(3,3) = 1680

5 Helpful 0 Interesting 1 Brilliant 0 Confused

If you view the grid as immobile, then 1680 is correct; however, if you are able to rotate the grid or view it from different angles, then permutations appear to repeat themselves depending on where you are standing. Since there are 4 different ways to view the square, divide 1680 by four, and the answer could be only 420.

A Former Brilliant Member - 2 years, 3 months ago
Info Web
Sep 25, 2020

First, out of 9 squares, 3 needs to be painted with any one color (say blue), in 9C3 ways.

Next, out of 6 squares left, 3 squares need to be painted with the other color (say red), in 6C3 ways.

Lastly, the last three squares needs to be painted with the last color (say green), in 3C3 ways.

So, total combinations = (9 C 3)(6 C 3)(3 C 3) = 1680

2 Helpful 0 Interesting 1 Brilliant 1 Confused
Syed Abdul Hafeez
May 30, 2021

Hello Everyone! How do I solve the below problem? Please note I changes the last part to "Each color 5 times"

How many different ways are there to color a 3×3 grid with green, red, and blue paints, using each color 5 times?

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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