document.write( "Question 1208703: A standard six-sided die is rolled $8$ times. You are told that among the rolls, there was one $1$, two $2$'s, four $3$'s, and one $4$. How many possible sequences of rolls could there have been? (For example, $2,$ $1,$ $3,$ $3,$ $3,$ $2,$ $3,$ $4$ is one possible sequence.) \n" ); document.write( "

Algebra.Com's Answer #847107 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "A standard six-sided die is rolled 8 times. You are told that among the rolls, there was one 1, two 2's, four 3's, and one 4.
\n" ); document.write( "How many possible sequences of rolls could there have been? (For example, 2, 1, 3, 3, 3, 2, 3, 4 is one possible sequence.)
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document.write( "There are 8 possible positions to place the given outcomes of 8 rolls.\r\n" );
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document.write( "The unique \"1\" can be placed to any of 8 positions.\r\n" );
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document.write( "The two 2's can be placed to any 2 of remaining 8-1 = 7 positions.\r\n" );
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document.write( "The four 3's can be placed to any 4 of  of remaining 7-2 = 5 positions.\r\n" );
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document.write( "The last \"4\" should be placed to unique remaining 5-4 = 1 position, with no choice.\r\n" );
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document.write( "The total number of different placements is   =  = 8*21*5*1 = 840.\r\n" );
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document.write( "ANSWER.   The total number of different placements is  840.\r\n" );
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\n" ); document.write( "\n" ); document.write( "We use combinations, since the order of outcomes inside each set of co-named outcomes does not matter.\r
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