document.write( "Question 1203598: How many permutations of the letters {A, B, C , D, E , F , G } are
\n" ); document.write( "there, such that
\n" ); document.write( "1 A precedes B
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\n" ); document.write( "4 C,D,E appear together in this order
\n" ); document.write( "5 A,B are appear together in this order, as do C , D \n" ); document.write( "
Algebra.Com's Answer #839276 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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\n" ); document.write( "How many permutations of the letters {A, B, C , D, E , F , G} are
\n" ); document.write( "there, such that
\n" ); document.write( "1 A precedes B
\n" ); document.write( "2 A precedes B, and C precedes D
\n" ); document.write( "3 A precedes B, and B precedes C
\n" ); document.write( "4 C,D,E appear together in this order
\n" ); document.write( "5 A,B \"highlight%28cross%28are%29%29\" appear together in this order, as do C,D
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\n" ); document.write( "\n" ); document.write( "        My reading and understanding is different from that of Edwin,\r
\n" ); document.write( "\n" ); document.write( "        so my solutions and my answers are different.\r
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\n" ); document.write( "\n" ); document.write( "(1)   A precedes B\r
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document.write( "     In English, \"A precedes B\" means that A goes/comes before B, but it is not necessary \r\n" );
document.write( "     that B goes/comes immediately after A: it may happen that there is / (there are) other letter / (letters) \r\n" );
document.write( "     after A, but before B.  In other words, \"A precedes B\" means that A goes/comes before B,\r\n" );
document.write( "     but A and B are not necessary adjacent (= not necessary appear altogether).\r\n" );
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document.write( "     Based on this understanding, there are 7! = 7*6*5*4*3*2*1 = 5040 permutations of 7 letters\r\n" );
document.write( "     {A, B, C , D, E , F , G}; of them, in exactly half cases, A precedes B. In other half, \r\n" );
document.write( "     in opposite, B precedes A.  To convince yourself that it is true, take any permutation\r\n" );
document.write( "     where A precedes B and make swap.\r\n" );
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document.write( "     So, the ANSWER to (1) is  \"7%21%2F2\" = \"5040%2F2\" = 2520.\r\n" );
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\n" ); document.write( "\n" ); document.write( "(2)   A precedes B, and C precedes D\r
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document.write( "     As we just learned from (a), there are  \"7%21%2F2\" = \"5040%2F2\" = 2520 permutations \r\n" );
document.write( "     of 7 letters {A, B, C , D, E , F , G}, such that A precedes B.\r\n" );
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document.write( "     Of them, in half cases C precedes D; in other half cases D precedes C.\r\n" );
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document.write( "     So, the ANSWER for (2) is  \"2520%2F2\" = 1260.  Or, which is the same, \"7%21%2F%282%2A2%29\" = \"7%21%2F4\".\r\n" );
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\n" ); document.write( "\n" ); document.write( "(3)   A precedes B, and B precedes C\r
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document.write( "     With any permutation of 7 letters  {A, B, C , D, E , F , G }, there are 3!-1 = 6-1 = 5 other permutations\r\n" );
document.write( "     where letters A, B and C are swapped in their positions.\r\n" );
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document.write( "     It gives you an idea, that with any favorable permutation, where A precedes B and B precedes C, there are 5 other\r\n" );
document.write( "     permutations with swapped A, B and C in their positions.\r\n" );
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document.write( "     So, the answer to (c) is  \"7%21%2F3%21\" = \"7%21%2F6\" = \"5040%2F6\" =  840.\r\n" );
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\n" ); document.write( "\n" ); document.write( "(4)   C,D,E appear together in this order.\r
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\n" ); document.write( "\n" ); document.write( "         For this part, I agree with Edwin.\r
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\n" ); document.write( "\n" ); document.write( "(5)   A,B appear together in this order, as do C , D\r
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document.write( "     In this part, formulation in the post is unclear (= does not sound harmonically *) and needs to be clarified.\r\n" );
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document.write( "     May be, you want to ask \"A,B appear together in the same order, as do C,D\"\r\n" );
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document.write( "     Then it is clear, but it is DIFFERENT from your original formulation.\r\n" );
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document.write( "     Having unclear formulation, I prefer do not touch this part.\r\n" );
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\n" ); document.write( "\n" ); document.write( "*)   In  Math,  if a problem does not sound harmonically,  it is a sure sign that it is formulated incorrectly.\r
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\n" ); document.write( "\n" ); document.write( "      Why ? - Because harmony is the hidden meaning of  Math.\r
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