SOLUTION: In how many ways can a picture be painted by using two or more of 7 different colors?

We will do each of those separately:



For each of those 7 colors, there are exactly 2 decisions we can make
concerning that color.  They are:
(1) Use it in the picture.
(2) Don't use it in the picture.

Suppose the 7 colors are R,O,Y,G,B,I,V.
There are 2 decisions we can make about color R.
For each of those, there are 2 decisions we can make about color O.
For each of those, there are 2 decisions we can make about color Y.
For each of those, there are 2 decisions we can make about color G.
For each of those, there are 2 decisions we can make about color B.
For each of those, there are 2 decisions we can make about color I.
For each of those, there are 2 decisions we can make about color V.

That's 2∙2∙2∙2∙2∙2∙2 = 27 = 128 ways.

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This one is easy because there is just 1 way to say "no" to all 7 colors,
and that 1 way is to say "no" to all of them.  (That is we paint the picture
in black and white, using NO colors).

That's 1 way.

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This one is also easy because there are 7 colors we can choose to
say "yes" to (saying "no" to each the other 6 colors)

That's 7 ways.

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So:






Edwin