document.write( "Question 877133: Cut a circle into 8 congruent parts.
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Algebra.Com's Answer #529213 by Edwin McCravy(20055)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Our approach will be this way:\r\n" );
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document.write( "\"%28matrix%288%2C1%2C%0D%0ANumber%2C+of%2Cways%2Cthat%2C+use%2C+ALL%2C+FOUR%2C+colors%29%29\" \"%22%22=%22%22\" \"%22%22-%22%22\"\"%22%22-%22%22\"\r\n" );
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document.write( "First we calculate the number of ways if we had no restriction\r\n" );
document.write( "on the number of colors we must use.  That is, we calculate as\r\n" );
document.write( "though we didn't have the restriction \"you must use all 4 colors\".\r\n" );
document.write( "That is, to begin with, we calculate the number of ways using 4 \r\n" );
document.write( "or fewer colors.\r\n" );
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document.write( "Choose the color for the 1st part 4 ways.\r\n" );
document.write( "Choose the color for the 2nd part 3 ways.\r\n" );
document.write( "Choose the color for the 3rd part 3 ways.\r\n" );
document.write( "Choose the color for the 4th part 3 ways.\r\n" );
document.write( "Choose the color for the 5th part 3 ways.\r\n" );
document.write( "Choose the color for the 6th part 3 ways.\r\n" );
document.write( "Choose the color for the 7th part 3 ways.\r\n" );
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document.write( "That's 4×3×3×3×3×3×3 = 4×36 = 2916 ways \r\n" );
document.write( "that we can use 4 or fewer colors and have no adjacent\r\n" );
document.write( "parts with the same color.\r\n" );
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document.write( "From the 2916 we will subtract:\r\n" );
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document.write( "A. the number that use exactly 3 of the 4 colors, and\r\n" );
document.write( "B. the number that use exactly 2 of the 4 colors.\r\n" );
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document.write( "A. We'll calculate the number of ways that use exactly  \r\n" );
document.write( "3 of the 4 colors.\r\n" );
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document.write( "We'll calculate the number of ways that use exact 3 SPECIFIED \r\n" );
document.write( "colors.  (Then we'll multiply that by C(4,3) ways to choose \r\n" );
document.write( "the 3 specified colors.)\r\n" );
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document.write( "We begin this by first calculating the number of ways that\r\n" );
document.write( "use 3 OR 2 colors. \r\n" );
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document.write( "Choose the color for the 1st part 3 ways.\r\n" );
document.write( "Choose the color for the 2nd part 2 ways.\r\n" );
document.write( "Choose the color for the 3rd part 2 ways.\r\n" );
document.write( "Choose the color for the 4th part 2 ways.\r\n" );
document.write( "Choose the color for the 5th part 2 ways.\r\n" );
document.write( "Choose the color for the 6th part 2 ways.\r\n" );
document.write( "Choose the color for the 7th part 2 ways.\r\n" );
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document.write( "That's 3×2×2×2×2×2×2 = 3×26 = 192 ways that use\r\n" );
document.write( "3 or fewer of 3 specified colors.\r\n" );
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document.write( "From that we must calculate and subtract the number \r\n" );
document.write( "of ways that use 2 of those specifed 3 colors.  There \r\n" );
document.write( "are only 2 such types:  ABABABA and BABABAB, where A and\r\n" );
document.write( "B represent different colors.  But since we have 3\r\n" );
document.write( "specified colors, we can choose the 2 colors any of \r\n" );
document.write( "C(3,2) or 3 ways. So that's 6 ways we must subtract \r\n" );
document.write( "from the 192.\r\n" );
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document.write( "So there are 192-6 = 186 ways to have exactly 3 \r\n" );
document.write( "specified colors.\r\n" );
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document.write( "But there are C(4,3) = 4 ways to chose the 3 specified\r\n" );
document.write( "colors.  So there are 4×186 = 744 ways that use exactly\r\n" );
document.write( "3 of the four colors.\r\n" );
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document.write( "B. Finally we must subtract the number which have exactly\r\n" );
document.write( "2 of the 4 colors.\r\n" );
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document.write( "We have already shown that there are only 2 types which\r\n" );
document.write( "use exactly 2 colors, ABABABA and BABABAB.  And there\r\n" );
document.write( "are C(4,2) = 6 ways to choose the 2 colors for them.\r\n" );
document.write( "So that's 6×2 or 12 ways to use exactly 2 colors.     \r\n" );
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document.write( "Answer: 2916-744-12 = 2160.\r\n" );
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document.write( "Edwin

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