document.write( "Question 1140059: （1）in how many ways can 4 boys and 3 girls be arrange in a line if the 3 girls are to be seperated
\n" ); document.write( "（2）when the number of the ways permuting the letters \"HELL\" such that （A）2ls well always be together （B）the 2ls will always be apart. \n" ); document.write( "

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(1) I would start by numbering the positions in the line as positions 1 through $\"3%2B4=7\"$ .
\n" ); document.write( "Then I would try to decide what positions to give the girls. If position 1 would go to a girl, I cannot place a girl in position 2, but a second girl could take position 3, and then maybe I can give position 5 to a the remaining girl. That would be a $\"1-3-5\"$ set of positions.
\n" ); document.write( "Other choices for position sets for the girls are:
\n" ); document.write( " $\"1-3-6\"$ , $\"1-3-7\"$ , $\"1-4-6\"$ , $\"1-4-7\"$ , $\"1-5-7\"$ ,
\n" ); document.write( " $\"2-4-6\"$ , $\"2-4-7\"$ , $\"2-5-7\"$ ,
\n" ); document.write( "and $\"3-5-7\"$ .
\n" ); document.write( "That is $\"10\"$ different sets of positions that could be reserved for the girls.
\n" ); document.write( "For each of those $\"10\"$ different options, I can still decide which girl goes first, second, and third. That gives us another set of $\"3%21=6\"$ girl ordering options (independently of the position arrangement), and
\n" ); document.write( " $\"10%2A6=60\"$ different ways arrange the girls.
\n" ); document.write( "That still leaves us the choice of how to arrange the 4 boys in the 4 empty positions. There are $\"4%21=24\"$ different ordering ways to arrange the boys.
\n" ); document.write( "That gives us a total of $\"10%2A6%2A24=highlight%281440%29\"$ different ways to arrange 4 boys and 3 girls in a line if the 3 girls are to be separated.
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\n" ); document.write( "(2) (A) Assuming which L appears first does not matter (because maybe we cannot tell them apart), there would be $\"3%21=2%2A3=highlight%286%29\"$ ways to arrange 3 tiles labelled H, E. and LL:
\n" ); document.write( "H E LL, H LL E, E H LL, E LL H, LL E H, and LL H E.
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\n" ); document.write( "(2) (B) To get the two L's separated we can have them either as the first and third letter, or as the second and fourth letter. That gives us two different options. That is $\"2\"$ choices for placement of the L's.
\n" ); document.write( "For each of those options we can choose to either have the H before the E, or the E before the H. That is $\"2\"$ choices independent of the L-placement options.
\n" ); document.write( "So assuming that we cannot tell the L's apart, there would be $\"2%2A2=highlight%284%29\"$ \"words\" we could form:
\n" ); document.write( "LHLE, LELH, HLEL, and ELHL .
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\n" ); document.write( "(2) NOTE: If we could tell apart the L's, the number of arrangements would be double, because either one or the other L could go first.
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