Grid Coloring

Logic Level 2

Find the number to color a 3 by 3 grid using 6 distinct colors, such that at least one cell of each row is colored. A coloring is considered different if it contains different color in at least one cell. You must use all 6 colors, no color can be used more than once.

Here are some possible valid colorings(all of these are different ways):


The answer is 58320.

Jaber Ahmed by Jaber Ahmed , 18, Bangladesh

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2 solutions

Jaber Ahmed
May 4, 2021

First let's select cells to color. This can be done in 9C6-3 ways (there are three cases where a row is empty). Now arrange colors in these 6 cells in 6! way. So answer is (9C6-3)*6! = 58320

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Shadman Tabib
May 4, 2021

(9!/3!)-6!*3

Having 6 different colors and 3 same color(blank) can be arranged in 9!/3! ways...Then we have to deduct the number of permutations which bear 3 blank colors at each row.And it will be 6! * 3(as there are 3 rows).. Finally we get (9!/3!)-(6!*3) That's it!!

0 Helpful 0 Interesting 3 Brilliant 1 Confused

I have found another solution...
There are 2 cases... The rows will be 1,2,3(colored) or 2,2,2(colored) configuration
1st case:6C1 * 5C2 * 3C3 * 3P1 * 3P2 * 3P3 * 3!
2nd case:6C2 * 4C2 * 2C2 * 3P2 * 3P2 * 3P2
By adding these-58320



Azim Sikder - 1 month, 1 week ago

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this was the intended solution tbh, doing casework as contraint is small :D

Jaber Ahmed - 1 month, 1 week ago

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