Question 1143762
In how many ways can a picture be painted by using two or more of 7 different
colors?<pre>
{{{(matrix(5,1,
matrix(1,3,number,of,ways),
matrix(1,3,the,picture,can),     
matrix(1,3,be,painted,using),
matrix(1,5,TWO,OR,MORE,OF,THE),
matrix(1,3,7,different,colors) ))}}}{{{matrix(5,1,"","",""="","","")}}}{{{(matrix(5,1,
matrix(1,3,number,of,ways),
matrix(1,3,the,picture,can),     
matrix(1,3,be,painted,using),
matrix(1,4,ANY,NUMBER,OF,THE),
matrix(1,3,7,different,colors) ))}}}{{{matrix(5,1,"","",""-"","","")}}}{{{(matrix(5,1,
matrix(1,3,number,of,ways),
matrix(1,3,the,picture,can),     
matrix(1,3,be,painted,using),
matrix(1,3,NONE,OF,THE),
matrix(1,3,7,different,colors) ))}}}{{{matrix(5,1,"","",""-"","","")}}}{{{(matrix(5,1,
matrix(1,3,number,of,ways),
matrix(1,3,the,picture,can),     
matrix(1,3,be,painted,using),
matrix(1,4,ONLY,ONE,OF,THE),
matrix(1,3,7,different,colors) ))}}}

We will do each of those separately:

{{{(matrix(5,1,
matrix(1,3,number,of,ways),
matrix(1,3,the,picture,can),     
matrix(1,3,be,painted,using),
matrix(1,4,ANY,NUMBER,OF,THE),
matrix(1,3,7,different,colors) ))}}}

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 = 2<sup>7</sup> = 128 ways.

------------------------------------------------

{{{(matrix(5,1,
matrix(1,3,number,of,ways),
matrix(1,3,the,picture,can),     
matrix(1,3,be,painted,using),
matrix(1,3,NONE,OF,THE),
matrix(1,3,7,different,colors) ))}}}

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.

------------------------------------------------

{{{(matrix(5,1,
matrix(1,3,number,of,ways),
matrix(1,3,the,picture,can),     
matrix(1,3,be,painted,using),
matrix(1,4,ONLY,ONE,OF,THE),
matrix(1,3,7,different,colors) ))}}}

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.

------------------------------------------------

So:

{{{(matrix(5,1,
matrix(1,3,number,of,ways),
matrix(1,3,the,picture,can),     
matrix(1,3,be,painted,using),
matrix(1,5,TWO,OR,MORE,OF,THE),
matrix(1,3,7,different,colors) ))}}}{{{matrix(5,1,"","",""="","","")}}}{{{(matrix(5,1,
matrix(1,3,number,of,ways),
matrix(1,3,the,picture,can),     
matrix(1,3,be,painted,using),
matrix(1,4,ANY,NUMBER,OF,THE),
matrix(1,3,7,different,colors) ))}}}{{{matrix(5,1,"","",""-"","","")}}}{{{(matrix(5,1,
matrix(1,3,number,of,ways),
matrix(1,3,the,picture,can),     
matrix(1,3,be,painted,using),
matrix(1,3,NONE,OF,THE),
matrix(1,3,7,different,colors) ))}}}{{{matrix(5,1,"","",""-"","","")}}}{{{(matrix(5,1,
matrix(1,3,number,of,ways),
matrix(1,3,the,picture,can),     
matrix(1,3,be,painted,using),
matrix(1,4,ONLY,ONE,OF,THE),
matrix(1,3,7,different,colors) ))}}}


{{{(matrix(5,1,
matrix(1,3,number,of,ways),
matrix(1,3,the,picture,can),     
matrix(1,3,be,painted,using),
matrix(1,5,TWO,OR,MORE,OF,THE),
matrix(1,3,7,different,colors) ))}}}{{{matrix(5,1,"","",""="","","")}}}{{{matrix(5,1,
"","",128,"","")}}}{{{matrix(5,1,"","",""-"","","")}}}{{{matrix(5,1,
"","",1,"","")}}}{{{matrix(5,1,"","",""-"","","")}}}{{{matrix(5,1,
"","",7,"","")}}}{{{matrix(5,1,"","",""="","","")}}}{{{matrix(5,1,
"","",matrix(1,2,120,ways),"","")}}}

Edwin</pre>