document.write( "Question 1203450: There are six available colors. find the number of ways to color the unit cells of a 2 x 2 grid such that no two adjacent cells are of the same color. solve the problem in two ways: by considering the number of colors used and by coloring the squares one after another. \n" ); document.write( "

Algebra.Com's Answer #839027 by ikleyn(52798) $\"\"$ $\"About$

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\n" ); document.write( "There are six available colors. find the number of ways to color the unit cells of a 2 x 2 grid
\n" ); document.write( "such that no two adjacent cells are of the same color. solve the problem in two ways:
\n" ); document.write( "by considering the number of colors used and by coloring the squares one after another.
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\n" ); document.write( "\n" ); document.write( " I will solve as my common sense dictates me, without even looking on instruction.\r
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document.write( "Imagine that the cells of the 2x2 grid are denoted in this way\r\n" );
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document.write( "        A1  A2\r\n" );
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document.write( "        B1  B2\r\n" );
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document.write( "There are 2 basic cases: (a) A2 and B1 are of different colors (and different from A1 and B2), \r\n" );
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document.write( "          and\r\n" );
document.write( "                         (b) A2 and B1 are of the same colors (and different from A1 and B2)\r\n" );
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document.write( "                         Consider case (a) first.\r\n" );
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document.write( "(a)  For A1, I can use any of 6 colors;\r\n" );
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document.write( "     then for A2 I can use any of remaining 5 colors;\r\n" );
document.write( "          for B1 I can use any of remaining 4 colors;\r\n" );
document.write( "          for B2 I can use (and I must use) one of remaining 3 colors OR the same color as A1.\r\n" );
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document.write( "     In all, I have 6*5*4*(3+1) = 6*5*4*4 = 480 different colorings in case (a).\r\n" );
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document.write( "                          Consider case (b) next.\r\n" );
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document.write( "(b)  For A1, I can use any of 6 colors;\r\n" );
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document.write( "     then for A2 I can use any of remaining 5 colors;\r\n" );
document.write( "          for B1 I use the same color as A2, so I have no choice;\r\n" );
document.write( "          for B2 I can use (and I must use) one of remaining 4 colors OR the same color as A1.\r\n" );
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document.write( "     In all, I have 6*5*1*(4+1) = 6*5*1*5 = 150 different colorings in case (b).\r\n" );
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document.write( "Cases (a) and (b) are DISJOINT, so the ANSWER to the problems' question is this sum\r\n" );
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document.write( "     480 + 150 = 630.\r\n" );
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document.write( "ANSWER.  630 different colorings.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "Do you see, this problem is nice and very good itself as an educational and/or
\n" ); document.write( "entertainment problem, but the attached instruction ruins the whole picture.\r
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\n" ); document.write( "\n" ); document.write( "For a student, who is able to solve this problem, this instruction is not needed.
\n" ); document.write( "For a student, who can not solve it, this instruction will not help. \r
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