document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=252789>Question 252789</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are five balls that must be played in a game. Two of the balls are signed<BR>\n" );
document.write( "balls and must be played either first or last. Each ball can only be played once.<BR>\n" );
document.write( "How many possible orderings (permutations) can the balls be played?\r<BR>\n" );
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document.write( "A) 2 B) 12  C) 60  D) 120  E) 3125 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #184858 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=184858\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> each signed ball have to be played in the first or last game so they are fixed at 2 games, one for each.\r<BR>\n" );
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document.write( "the other 3 balls can be in games 2,3,4 in any order.\r<BR>\n" );
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document.write( "all balls can be played in one game only.\r<BR>\n" );
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document.write( "the number of permutations for the 3 unsigned balls is 3! = 3*2*1 = 6.\r<BR>\n" );
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document.write( "the two signed balls can be played in the first or last position only.\r<BR>\n" );
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document.write( "the number  of permutations for the signed balls is 2!.\r<BR>\n" );
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document.write( "the total order of permutations is 2 * 6 = 12.\r<BR>\n" );
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document.write( "that's 2 permutations for the signed balls times 6 permutations for the unsigned balls.\r<BR>\n" );
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document.write( "here's how it works.\r<BR>\n" );
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document.write( "let a,e be the signed balls.<BR>\n" );
document.write( "let b,c,d be the unsigned balls.\r<BR>\n" );
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document.write( "your possible permutations for games 1,2,3,4,5 are:\r<BR>\n" );
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document.write( "a,b,c,d,e<BR>\n" );
document.write( "a,b,d,c,e<BR>\n" );
document.write( "a,c,b,d,e<BR>\n" );
document.write( "a,c,d,b,e<BR>\n" );
document.write( "a,d,b,c,e<BR>\n" );
document.write( "a,d,c,b,e\r<BR>\n" );
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document.write( "e,b,c,d,a<BR>\n" );
document.write( "e,b,d,c,a<BR>\n" );
document.write( "e,c,b,d,a<BR>\n" );
document.write( "e,c,d,b,a<BR>\n" );
document.write( "e,d,b,c,a<BR>\n" );
document.write( "e,d,c,b,a\r<BR>\n" );
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document.write( " </SPAN>\n" );
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