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# In how many different ways you can paint your fingernails with 3 colors, but only 2 colors on one hand?

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You have 3 different colors of nail polish. You can paint one fingernail with one color, one hand with no more than two colors. In how many different ways you can paint your fingernails? The answer in the book doesn't help at all :) So it would be very helpful to understand the solution not just see the answer. And the answer is (totally surprising):

$${[{3! \over 2!(3-2)!}\times({2}^{5}-1)]}^{2}=8649$$

May 5, 2020

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I came up with this (wrong) answer:

Let's consider my first hand. I have two colors to choose from. So my first fingernail I can paint with two different colors, same with my second, third, fourth, and fifth fingernail: 2 x 2 x 2 x 2 x 2. But there are 3 colors to choose form. So I have to multiply 2^5 by combination, where n = 2 and k = 2. I have 2 hands so I have to raise it to the power of 2. But why there is minus 1 in the solution. It's just k*****g me :(

May 5, 2020