document.write( "Question 1149724: The letters of ENTERTAINMENT are arranged in a row. Find the probability that the letters E are apart.
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Algebra.Com's Answer #771140 by Edwin McCravy(20055)\"\" \"About 
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document.write( "ENTERTAINMENT\r\n" );
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document.write( "In alphabetical order, the letters are A,E,E,E,I,M,N,N,N,R,T,T,T.\r\n" );
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document.write( "First we find the number of distinguishable arrangements, regardless of\r\n" );
document.write( "whether the three E's are apart or not.\r\n" );
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document.write( "That's 13 things with 3 indistinguishable E's, 3 indistinguishable N's, and 3\r\n" );
document.write( "indistinguishable T's,\r\n" );
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document.write( "That's \"13%21%2F%283%213%213%21%29=28828800\"\r\n" );
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document.write( "From that we will subtract the ones with some E's together. They consist of\r\n" );
document.write( "two cases:\r\n" );
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document.write( "Case 1. the number of arrangements that have all three E's together, like EEE.\r\n" );
document.write( "That's the number of arrangements of these 11 \"things\" with 3\r\n" );
document.write( "indistinguishable N's and 3 indistinguishable T's.\r\n" );
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document.write( "A,EEE,I,M,N,N,N,R,T,T,T.\r\n" );
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document.write( "That's \"11%21%2F%283%213%21%29=1108800\" ways.\r\n" );
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document.write( "and\r\n" );
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document.write( "Case 2. the number of arrangements that have EE together and the E apart from it.\r\n" );
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document.write( "First we find the number of arrangements without any E's.  That's the number\r\n" );
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document.write( "of arrangements of these 10 things, which have 3 indistinguishable N's and 3\r\n" );
document.write( "indistinguishable T's:\r\n" );
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document.write( "A,I,M,N,N,N,R,T,T,T.\r\n" );
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document.write( "That's \"10%21%2F%283%213%21%29=100800\"\r\n" );
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document.write( "Then we'll insert an EE and an E among them so that we don't put them\r\n" );
document.write( "together, avoiding counting again the ones with EEE from case 1.\r\n" );
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document.write( "To do that, we now look at a random arrangement from the 100800 with no E's at\r\n" );
document.write( "all, say this one:\r\n" );
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document.write( "T,R,N,A,I,T,N,N,M,T\r\n" );
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document.write( "We put 8 spaces between the letters, 1 space in the beginning, and 1 space at\r\n" );
document.write( "the end. That's 10 spaces, and we'll put an E in one of them and EE in another\r\n" );
document.write( "one.  That way the E and the EE won't be together.\r\n" );
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document.write( "_T_R_N_A_I_N_N_M_T_\r\n" );
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document.write( "We can choose the space to put the single E in in 10 ways.\r\n" );
document.write( "That leaves 9 spaces to put the double EE in.\r\n" );
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document.write( "So for each of the 100800 ways we can insert the E and the EE in 10∙9 or 90 \r\n" );
document.write( "ways.  That's 100800∙90 = 9072000 ways to have EE separate from E.\r\n" );
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document.write( "So the total number that we must subtract from 28828800 is 1108800 from case 1\r\n" );
document.write( "and 9072000 from case 2.\r\n" );
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document.write( "28828800-1108800-9072000 = 15348000 ways.\r\n" );
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document.write( "Edwin
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