How many combos: 9-pan eyeshadow palette

Hello everyone! Welcome to something that's been a little while in the making. I've been curious for a while - how many unique color combinations are in an eyeshadow palette? This is the question we're exploring today.
Obviously, the answer to this question differs a little on how many pans are in an eyeshadow palette to begin with - there will be a different number of color combination in a Wet n Wild Quad than there will be in a 35-pan Morphe palette, just because the Morphe one has almost 9x the number of pans.
Because I'm doing Pan That Palette this year with the Too Faced Peanut Butter and Jelly Palette this year, I decided to see how many different 1-color, 2-color, 3-color, and 4-color combinations there would be in a nine-pan palette.

A few disclaimers before I start this. First, this doesn't account for the number of potential looks in a palette. For example, with the PBJ Palette, I'm counting the combination of Extra Creamy and Nuts About U as one look/ combination of colors. You could hypothetically wear Nuts About U on the lid and Extra Creamy to blend out the edges; you could also use Extra Creamy on the lid and Nuts About U on the crease. The methods I used just account for the number of combinations, not necessarily the number of looks.
Second, there may be a bit of human error here. Because I was trying to avoid repeat numbers, and it gets monotonous after a little while, I may have accidentally missed something. I tried to back and double-check, but do be aware I'm human.

With all that said, let's get into the numbers.

In a nine-pan palette, there are nine one-color looks. This is simple - there are nine pans, so there are nine one-color looks.
There are 36 two-color looks. The way I did this was to just count off the number of combinations 1-2, 1-3, 1-4, and then 2-3, 2-4, etc.
According to my calculations, there are 84 three-color combinations. This is a similar method to the two-color combinations: 1-2-3, 1-2-4, 1-2-5, 1-4-5, 2-5-6, all the way down to 7-8-9.
According to my calculations, there are 126 four-color combinations. The methods are exactly the same as the others.

That means that, altogether, there are 255 unique color combinations in a 9-pan palette. Again, this is not necessarily the number of looks. This doesn't account for the color placement or anything like that. There's probably more than just 255 looks in a palette with 9 pans, but this is the color combinations we're talking about.

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