document.write( "Question 829874: All letters of the word \"EAMCET\"are arranged in all possible ways.The number of such arrangement in which two vowels are not adjacent to each other. \n" ); document.write( "
Algebra.Com's Answer #500216 by Edwin McCravy(20055)\"\" \"About 
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document.write( "First we calculate the number of distinguishable permutations of EAMCET.\r\n" );
document.write( "The 2 E's are not distinguishable so that number is \r\n" );
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document.write( "     \"6%21%2F2%21\" = 360\r\n" );
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document.write( "Then from those 360 we will subtract \r\n" );
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document.write( "(1) the number of ways all 3 vowels come together.\r\n" );
document.write( "and\r\n" );
document.write( "(2) the number of ways exactly 2 vowels come together.\r\n" );
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document.write( "We calculate (1), the number of ways all 3 vowels come together\r\n" );
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document.write( "I will use the notation N{i,j,k,...} to indicate the number of permutations\r\n" );
document.write( "of the elements between the braces.\r\n" );
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document.write( "N{EEA,M,C,T} = N(EAE,M,C,T} = N{AEE,M,C,T} = 4! = 24\r\n" );
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document.write( "So the number of ways all three vowels can come together is 3·4! = 72\r\n" );
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document.write( "Next we calculate (2), the number of ways exactly 2 vowels come together\r\n" );
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document.write( "N{AE,E,M,C,T} - N{AEE,M,C,T} - N{EAE,M,C,T} = \r\n" );
document.write( "N{EA,E,M,C,T} - N{EAE,M,C,T} - N{EEA,M,C,T} = \r\n" );
document.write( "N{EE,A,M,C,T} - N{EEA,M,C,T} - N{AEE,M,C,T} = 5!-4!-4! = 5!-2*4! = 72\r\n" );
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document.write( "So the number of ways exactly 2 vowels come together = 3(5!-2*4!) = 216\r\n" );
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document.write( "Answer: \"6%21%2F2%21\"\"%22%22-%22%22\"\"3%2A4%21\"\"%22%22-%22%22\"\"3%285%21-2%2A4%21%29\" = \"360+-+72+-+216+=+72\"\r\n" );
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document.write( "As a check, with the aid of my computer, here are all 72 such permutations\r\n" );
document.write( "in 8 rows of 9 each:\r\n" );
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document.write( "EMECAT, EMECTA, EMETAC, EMETCA, EMACET, EMACTE, EMATEC, EMATCE, EMCETA,\r\n" );
document.write( "EMCATE, EMTECA, EMTACE, ECEMAT, ECEMTA, ECETAM, ECETMA, ECAMET, ECAMTE,\r\n" );
document.write( "ECATEM, ECATME, ECMETA, ECMATE, ECTEMA, ECTAME, ETEMAC, ETEMCA, ETECAM,\r\n" );
document.write( "ETECMA, ETAMEC, ETAMCE, ETACEM, ETACME, ETMECA, ETMACE, ETCEMA, ETCAME,\r\n" );
document.write( "AMECET, AMECTE, AMETEC, AMETCE, AMCETE, AMTECE, ACEMET, ACEMTE, ACETEM,\r\n" );
document.write( "ACETME, ACMETE, ACTEME, ATEMEC, ATEMCE, ATECEM, ATECME, ATMECE, ATCEME,\r\n" );
document.write( "MECETA, MECATE, METECA, METACE, MACETE, MATECE, CEMETA, CEMATE, CETEMA,\r\n" );
document.write( "CETAME, CAMETE, CATEME, TEMECA, TEMACE, TECEMA, TECAME, TAMECE, TACEME.\r\n" );
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document.write( "Edwin
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