document.write( "Question 206861: Roman has a bag containing marbles of four different colours. How many ways can he place four marbles in a row so that no two marbles adjacent to each other are of the same colour?\r
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Algebra.Com's Answer #156398 by Edwin McCravy(20060) $\"\"$ $\"About$

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Roman has a bag containing marbles of four different colours. How many ways can he place four marbles in a row so that no two marbles adjacent to each other are of the same colour?\r
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document.write( "1.  Cases where all 4 are different colors:\r\n" );
document.write( "  \r\n" );
document.write( "    Choose a color for the 1st marble 4 ways.  That's 4 ways.\r\n" );
document.write( "    Choose a color for the 2nd marble 3 ways.  That's 4*3 ways.\r\n" );
document.write( "    Choose a color for the 3rd marble 2 ways.  That's 4*3*2 ways.\r\n" );
document.write( "    Choose a color for the 4th marble 1 way.   That's 4*3*2*1 ways.\r\n" );
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document.write( "    These cases account for 24 ways.\r\n" );
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document.write( "2. Cases where the 1st and 3rd are the same color and the 2nd and 4th\r\n" );
document.write( "   are the same color.\r\n" );
document.write( "\r\n" );
document.write( "   Choose a color for the 1st and 3rd.  That's 4 ways.\r\n" );
document.write( "   Choose a color for the 2nd and 4th.  That's 4*3 ways.\r\n" );
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document.write( "   These cases accounts for 12 ways.\r\n" );
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document.write( "3. Cases where the 1st and 3rd are the same color and the 2nd and 4th\r\n" );
document.write( "   are different colors.    \r\n" );
document.write( "\r\n" );
document.write( "   Choose the color for the 1st and 3rd.  That's 4 ways\r\n" );
document.write( "   Choose the color for the 2nd,  That's 4*3 ways.\r\n" );
document.write( "   Choose the color for the 4th.  That's 4*3*2.\r\n" );
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document.write( "   These cases accounts for 24 ways.\r\n" );
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document.write( "4. Cases where the 2nd and 4th are the same color and the 1st and 3rd\r\n" );
document.write( "   are different colors.    \r\n" );
document.write( "\r\n" );
document.write( "   Choose the color for the 2nd and 4th.  That's 4 ways\r\n" );
document.write( "   Choose the color for the 1st,  That's 4*3 ways.\r\n" );
document.write( "   Choose the color for the 3rd.  That's 4*3*2.\r\n" );
document.write( "\r\n" );
document.write( "   These cases accounts for 24 ways.\r\n" );
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document.write( "5. Cases where the 1st and 4th are the same color and the 2nd and 3rd\r\n" );
document.write( "   are different colors. \r\n" );
document.write( "\r\n" );
document.write( "   Choose the color for the 1st and 4th.  That's 4 ways\r\n" );
document.write( "   Choose the color for the 2nd,  That's 4*3 ways.\r\n" );
document.write( "   Choose the color for the 3rd.  That's 4*3*2.\r\n" );
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document.write( "   These cases accounts for 24 ways.\r\n" );
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document.write( "Adding:\r\n" );
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document.write( "Case 1:  24\r\n" );
document.write( "Case 2:  12\r\n" );
document.write( "Case 3:  24\r\n" );
document.write( "Case 4:  24\r\n" );
document.write( "Case 5:  24\r\n" );
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document.write( "Total:  108\r\n" );
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document.write( "So the correct choice is (B) 108\r\n" );
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document.write( "Edwin

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