document.write( "Question 531425: How many distinguishable arrangements are there of the letters in \"arrangement\" that do not have the two \"r\"s adjacent? \n" ); document.write( "
Algebra.Com's Answer #350464 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "First we must calculate the number of distinguishable arrangements\r\n" );
document.write( "whether the R's come together or not, and then we must calculate the\r\n" );
document.write( "number of arrangements when they come together, and then subtract\r\n" );
document.write( "the second number from the first.\r\n" );
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document.write( "If it were \r\n" );
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document.write( "ARraNGEMenTS, (where you could tell A from a, R from r, and N from n,\r\n" );
document.write( "and E from e, the number of arrangements would be 12!\r\n" );
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document.write( "But since we cannot tell the difference between A and a, since they are both\r\n" );
document.write( "capital, we have to divide by 2! so as not to count, say, these two,\r\n" );
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document.write( "AaEeGMNnRrST and aAEeGMNnRrST\r\n" );
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document.write( "as two separate arrangements.\r\n" );
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document.write( "So we take out what we consider as A-duplications and we get \"12%21%2F2%21\"\r\n" );
document.write( "as the number of arrangements of AAEeGMNnRrST.\r\n" );
document.write( "---\r\n" );
document.write( "Also since we cannot tell the difference between E and e, since they are both\r\n" );
document.write( "capital, we have to divide by 2! again so as not to count, say, these two,\r\n" );
document.write( "AAEeGMNnRrST and AAeEGMNnRrST\r\n" );
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document.write( "as two separate arrangements.\r\n" );
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document.write( "So we take out what we consider as E-duplications and we get \"12%21%2F%282%212%21%29\"\r\n" );
document.write( "as the number of arrangements of AAEEGMNnRrST.\r\n" );
document.write( "---\r\n" );
document.write( "Also since we cannot tell the difference between N and n, since they are both\r\n" );
document.write( "capital, we have to divide by 2! again so as not to count, say, these two,\r\n" );
document.write( "AAEEGMNnRrST and AAEEGMnNRrST\r\n" );
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document.write( "as two separate arrangements.\r\n" );
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document.write( "So we take out what we consider as N-duplications and we get \"12%21%2F%282%212%212%21%29\"\r\n" );
document.write( "as the number of arrangements of AAEEGMNNRrST.\r\n" );
document.write( "---\r\n" );
document.write( "Also since we cannot tell the difference between R and r, since they are both\r\n" );
document.write( "capital, we have to divide by 2! again so as not to count, say, these two,\r\n" );
document.write( "AAEEGMNNRrST and AAEEGMNNrRST\r\n" );
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document.write( "as two separate arrangements.\r\n" );
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document.write( "So we take out what we consider as R-duplications and we get \"12%21%2F%282%212%212%212%21%29\" \r\n" );
document.write( "as the number of arrangements of AAEEGMNNRRST.  \r\n" );
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document.write( "That's \"479001600%2F%282%2A2%2A2%2A2%29\" = \"479001600%2F16\" = 29937600  \r\n" );
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document.write( "-----\r\n" );
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document.write( "Now we must calculate the number of arrangements where the R's come together,\r\n" );
document.write( "so we can subtract them from the above number.\r\n" );
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document.write( "If the R's come together, we are finding the number of arrangements\r\n" );
document.write( " of the 11 things A[RR]ANGEMENTS, where the [RR] is considered as\r\n" );
document.write( "if it were one single letter instead of two letters.\r\n" );
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document.write( "--------- \r\n" );
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document.write( "Similar to the first, if it were \r\n" );
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document.write( "A[RR]aNGEMenTS, (where you could tell A from a, R from r, and N from n,\r\n" );
document.write( "and E from e, the number of arrangements would be 11!\r\n" );
document.write( "\r\n" );
document.write( "But since we cannot tell the difference between A and a, since they are both\r\n" );
document.write( "capital, we have to divide by 2! so as not to count, say, these two,\r\n" );
document.write( "\r\n" );
document.write( "AaEeGMNn[RR]ST and aAEeGMNn[RR]ST\r\n" );
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document.write( "as two separate arrangements.\r\n" );
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document.write( "So we take out what we consider as A-duplications and we get \"11%21%2F2%21\"\r\n" );
document.write( "as the number of arrangements of AAEeGMNnRRST.\r\n" );
document.write( "---\r\n" );
document.write( "Also since we cannot tell the difference between E and e, since they are both\r\n" );
document.write( "capital, we have to divide by 2! again so as not to count, say, these two,\r\n" );
document.write( "AAEeGMNn[RR]ST and AAeEGMNn[RR]ST\r\n" );
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document.write( "as two separate arrangements.\r\n" );
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document.write( "So we take out what we consider as E-duplications and we get \"11%21%2F%282%212%21%29\"\r\n" );
document.write( "as the number of arrangements of AAEEGMNn[RR]ST.\r\n" );
document.write( "---\r\n" );
document.write( "Also since we cannot tell the difference between N and n, since they are both\r\n" );
document.write( "capital, we have to divide by 2! again so as not to count, say, these two,\r\n" );
document.write( "AAEEGMNn[RR]ST and AAEEGMnN[RR]ST\r\n" );
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document.write( "as two separate arrangements.\r\n" );
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document.write( "So we take out what we consider as N-duplications and we get \"11%21%2F%282%212%212%21%29\"\r\n" );
document.write( "as the number of arrangements of AAEEGMNN[RR]ST.\r\n" );
document.write( "  \r\n" );
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document.write( "That's \"39916800%2F%282%2A2%2A2%29\" = \"39916800%2F16\" = 4989600\r\n" );
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document.write( "--------------------\r\n" );
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document.write( "So we subtract to get the final answer:\r\n" );
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document.write( "29937600 - 4989600 = 2948000\r\n" );
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document.write( "And that's the number of distinguishable arrangements of ARRANGEMENTS\r\n" );
document.write( "where the two R's are not together.\r\n" );
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document.write( "Edwin
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