Painting a 4x4

Probability Level 3

How many ways are there to paint a $4\times4$ grid with four colors such that each row or column has all four distinct colors?

The answer is 576.

1 solution

Geoff Pilling
May 12, 2017

Not sure if there is a more intuitive solution, but this is what I came up with.

There are 24 ways to pick the colors first row.

Once these are picked there are 6 ways to pick the remaining colors in the first column.

Then, for each of these there are 4 ways to color the remaining squares.

$24 \cdot 6 \cdot 4 = \boxed{576}$

@Geoff Pilling Could you please explain this....."Then, for each of these there are 4 ways to color the remaining squares."??

Aaghaz Mahajan - 3 years, 3 months ago

WLOG you can take the example in the picture. Once you have the first two rows there are two ways to pick the colors in the first column, BG or GB. WLOG let's pick B on top. Then you have 6 squares left.

For the third row, you can either pick RY or YR. WLOG lets pick R on top.

Now the rest of the squares are uniquely determined.

Geoff Pilling - 3 years, 3 months ago

Painting a 4x4

The answer is 576.

1 solution

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