document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201548>Question 1201548</A>: <I><SPAN STYLE=\"word-break: break-all;\"> if P(n,2)=110 find n </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #835983 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=835983\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red size=4>n = 11</font>\r<BR>\n" );
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document.write( "Work Shown:\r<BR>\n" );
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document.write( "P(n,r) = n*(n-1)*(n-2)*...*(n-r+1) is the permutation function\r<BR>\n" );
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document.write( "When r = 2,<BR>\n" );
document.write( "P(n,2) = n*(n-1)\r<BR>\n" );
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document.write( "So,<BR>\n" );
document.write( "P(n,2) = 110<BR>\n" );
document.write( "n(n-1) = 110<BR>\n" );
document.write( "n^2-n = 110<BR>\n" );
document.write( "n^2-n-110 = 0\r<BR>\n" );
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document.write( "We could use a trial-and-error approach to factor.<BR>\n" );
document.write( "But I find the most efficient pathway is to use the quadratic formula.<BR>\n" );
document.write( "Think of n as x<BR>\n" );
document.write( "We have x^2-x-110 = 0 in the form ax^2+bx+c = 0<BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = -1<BR>\n" );
document.write( "c = -110<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+11 ALIGN=MIDDLE  ALT=\"x+=+11\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-10 ALIGN=MIDDLE  ALT=\"x+=+-10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Your steps for the quadratic formula do not need to be this verbose.\r<BR>\n" );
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document.write( "The solutions x = 11 or x = -10 lead back to n = 11 or n = -10.\r<BR>\n" );
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document.write( "The permutation function only allows positive integers for n.<BR>\n" );
document.write( "Therefore, we eliminate n = -10 and circle <font color=red>n = 11 as the final answer</font>.\r<BR>\n" );
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document.write( "Check:<BR>\n" );
document.write( "P(n,2) = n*(n-1)<BR>\n" );
document.write( "P(11,2) = 11*(11-1)<BR>\n" );
document.write( "P(11,2) = 11*10<BR>\n" );
document.write( "P(11,2) = 110<BR>\n" );
document.write( "The answer is confirmed.\r<BR>\n" );
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document.write( "Real world application:<BR>\n" );
document.write( "There are 11 people running for the government positions \"treasurer\" and \"secretary\". This means order matters.<BR>\n" );
document.write( "There are 11*10 = 110 different permutations possible.<BR>\n" );
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