document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=843235>Question 843235</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A music class of 5 girls and 4 boys is having a recital. If each member is to perform once,how many ways can the program be arranged in each of the following cases? \r<BR>\n" );
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document.write( "(A) all girls must perform first. <BR>\n" );
document.write( "(B) A girl must perform first and a boy must perform last. <BR>\n" );
document.write( "(C)Elsa and Doug will perform first and Last. Respectively.<BR>\n" );
document.write( "(D) The entire program will alternate between boy and girl. <BR>\n" );
document.write( "(E)The First, Fifth, and ninth performer must be girls. \r<BR>\n" );
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document.write( "I think it is combination. <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #508111 by </B><A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><B>Edwin McCravy(20056)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Edwin+McCravy><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Edwin+McCravy><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=508111\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "It cannot be combinations, because these are ORDERINGS of performances\r\n" );
document.write( "Permutations are ORDERINGS of choices.  Combinations are CHOICES only.\r\n" );
document.write( "So if it were combinations, then the answer would be 1, which means that\r\n" );
document.write( "all 9 people are chosen to perform, regardless of what order they perform\r\n" );
document.write( "in.  Since the ORDER in which they perform is the only thing that matters\r\n" );
document.write( "in any of these problems, these are PERMUTATIONS, not combinations.\r\n" );
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document.write( "(A) all girls must perform first.<BR>\n" );
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document.write( "Choose one of 5 girls to perform 1st.  \r\n" );
document.write( "That's 5 ways to choose the first performer.\r\n" );
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document.write( "Choose one of the 4 remaining girls to perform 2nd.\r\n" );
document.write( "That's 5×4 ways to choose the order of the first 2 performers. \r\n" );
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document.write( "Choose one of the 3 remaining girls to perform 3rd.\r\n" );
document.write( "That's 5×4×3 ways to choose the order of the first 3 performers.\r\n" );
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document.write( "Choose one of the 2 remaining girls to perform 4th.\r\n" );
document.write( "That's 5×4×3×2 ways to choose the order of the first 4 performers.\r\n" );
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document.write( "Choose the 1 remaining girl to perform 5th.\r\n" );
document.write( "That's 5×4×3×2×1 ways to choose the order of the first 5 performers.\r\n" );
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document.write( "So far that 5! or 5×4×3×2×1 = 120 = P(5,5) permutations \r\n" );
document.write( "or WAYS TO ORDER the 5 girls, taking all 5 of them.\r\n" );
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document.write( "Choose one of 4 boys to perform 6th.  \r\n" );
document.write( "That's 5×4×3×2×1×4 ways to choose the order of the first 6 performers.\r\n" );
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document.write( "Choose one of the 3 remaining boys to perform 7th.\r\n" );
document.write( "That's 5×4×3×2×1×4×3 ways to choose the order of the first 7 performers. \r\n" );
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document.write( "Choose one of the 2 remaining boys to perform 8th.\r\n" );
document.write( "That's 5×4×3×2×1×4×3×2 ways to choose the order of the first 8 performers. \r\n" );
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document.write( "Choose the 1 remaining boy to perform 9th or last.\r\n" );
document.write( "That's 5×4×3×2×1×4×3×2×1 ways to choose the order of all 9 performers. \r\n" );
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document.write( "The short way to do this is P(5,5)×P(4,4) = (5×4×3×2×1)×(4×3×2×1) = \r\n" );
document.write( "120×24 = 2800. \r\n" );
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document.write( " (B) A girl must perform first and a boy must perform last.<BR>\n" );
document.write( "---------------------------------------<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "Choose one of 5 girls to perform 1st.  \r\n" );
document.write( "That's 5 ways to choose the first performer.\r\n" );
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document.write( "Choose one of 4 boys to perform last or 9th.  \r\n" );
document.write( "That's 5×4 ways to choose the 1st and 9th performers.\r\n" );
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document.write( "Choose one of the 7 remaining people to perform 2nd.\r\n" );
document.write( "That's 5×4×7 ways to choose the order of the 1st 2 and 9th performers. \r\n" );
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document.write( "Choose one of the 6 remaining people to perform 3rd.\r\n" );
document.write( "That's 5×4×7×6 ways to choose the order of the 1st 3 and 9th performers.\r\n" );
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document.write( "Choose one of the 5 remaining people to perform 4th.\r\n" );
document.write( "That's 5×4×7×6×5 ways to choose the order of the 1st 4 and 9th performers.\r\n" );
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document.write( "Choose one of the 4 remaining people to perform 5th.\r\n" );
document.write( "That's 5×4×7×6×5×4 ways to choose the order of the 1st 5 and 9th performers.\r\n" );
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document.write( "Choose one of the 3 remaining people to perform 6th.\r\n" );
document.write( "That's 5×4×7×6×5×4×3 ways to choose the order of the 1st 6 and 9th performers.\r\n" );
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document.write( "Choose one of the 2 remaining people to perform 7th.\r\n" );
document.write( "That's 5×4×7×6×5×4×3×2 ways to choose the order of the 1st 7 and 9th performers.\r\n" );
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document.write( "Choose the 1 remaining person to perform 8th.\r\n" );
document.write( "That's 5×4×7×6×5×4×3×2×1 ways to choose the order of the 1st 8 and 9th performers.  \r\n" );
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document.write( "So the answer is:  5×4×7×6×5×4×3×2×1 = 100800.\r\n" );
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document.write( "Short way:  Choose the 1st girl P(5,1), Choose the last boy P(4,1),\r\n" );
document.write( "Choose the orderings for the middle 7 P(7,7)\r\n" );
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document.write( "Answer: P(5,1)×P(4,1)×P(7,7) = 5×4×5040 = 100800  \r\n" );
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document.write( "(C)Elsa and Doug will perform first and Last. Respectively.<BR>\n" );
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document.write( "Choose Elsa P(1,1)=1 way.  Choose Doug P(1,1) = 1 way.  Choose the\r\n" );
document.write( "middle 7 performers P(7,7) ways.  \r\n" );
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document.write( "Answer: P(1,1)×P(1,1)×P(7,7) = 1×1×5040 = 5040 orderings.\r\n" );
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document.write( "(D) The entire program will alternate between boy and girl.<BR>\n" );
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document.write( "Since there are 5 girls the program must go GBGBGBGBG.  We choose\r\n" );
document.write( "the odd numbered performers as the girls in P(5,5) ways, and for\r\n" );
document.write( "each of those orderings for the girls, we choose\r\n" );
document.write( "the odd numbered performers as the boys in P(4,4) ways.\r\n" );
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document.write( "That's P(5,5)×P(4,4) = 120×24 = 2880 orderings. \r\n" );
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document.write( "(E)The First, Fifth, and ninth performer must be girls. <BR>\n" );
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document.write( "Choose the 1st, 5th, and 9th performers as P(5,3). Choose the\r\n" );
document.write( "orderings for the other performers from the 6 remaining people P(6,6).  \r\n" );
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document.write( "Answer P(5,3)×P(6,6) = (5×4×3)\r\n" );
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document.write( "Edwin</PRE> </SPAN>\n" );
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