document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1141509>Question 1141509</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this:\r<BR>\n" );
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document.write( "Find the fifth term in the expansion of (ab-1) ^20. Use the term formula. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #762075 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52788)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=762075\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
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document.write( "The binomial expansion is this formula\r\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2By%29%5En ALIGN=MIDDLE  ALT=\"%28x%2By%29%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5En ALIGN=MIDDLE  ALT=\"x%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5Bn%5D%5E1%2Ax%5E%28n-1%29%2Ay ALIGN=MIDDLE  ALT=\"C%5Bn%5D%5E1%2Ax%5E%28n-1%29%2Ay\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5Bn%5D%5E2%2Ax%5E%28n-2%29%2Ay%5E2 ALIGN=MIDDLE  ALT=\"C%5Bn%5D%5E2%2Ax%5E%28n-2%29%2Ay%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5Bn%5D%5E3%2Ax%5E%28n-3%29%2Ay%5E3 ALIGN=MIDDLE  ALT=\"C%5Bn%5D%5E3%2Ax%5E%28n-3%29%2Ay%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > + . . . + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5Bn%5D%5E%28n-1%29%2Ax%5E1%2Ay%5E%28n-1%29 ALIGN=MIDDLE  ALT=\"C%5Bn%5D%5E%28n-1%29%2Ax%5E1%2Ay%5E%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5En ALIGN=MIDDLE  ALT=\"y%5En\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r\n" );
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document.write( "In our case,  n = 20,  x = ab,  y = -1,  therefore, the common term of the binomial expansion in our case is  \r\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5B20%5D%5Ek%2A%28ab%29%5E%2820-k%29%2A%28-1%29%5Ek ALIGN=MIDDLE  ALT=\"C%5B20%5D%5Ek%2A%28ab%29%5E%2820-k%29%2A%28-1%29%5Ek\"   DISABLED_style= \"max-width:90%; height:auto;\" >, k = 0, 1, 2, 3, 4, 5, . . . \r\n" );
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document.write( "Then the 5-th term is at k = 4 \r\n" );
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document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%5B20%5D%5E4%2A%28ab%29%5E%2820-4%29%2A%28-1%29%5E4 ALIGN=MIDDLE  ALT=\"C%5B20%5D%5E4%2A%28ab%29%5E%2820-4%29%2A%28-1%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%2820%2A19%2A18%2A17%29%2F%281%2A2%2A3%2A4%29%29%2A%28ab%29%5E16 ALIGN=MIDDLE  ALT=\"%28%2820%2A19%2A18%2A17%29%2F%281%2A2%2A3%2A4%29%29%2A%28ab%29%5E16\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4845%2Aa%5E16%2Ab%5E16 ALIGN=MIDDLE  ALT=\"4845%2Aa%5E16%2Ab%5E16\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r\n" );
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document.write( "If you want to see other similar solved problems and to learn the subject wider and deeper, &nbsp;look into the lesson\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/Permutations/Solved-problems-on-binomial-coefficients.lesson>Solved problems on binomial coefficients</A> \r<BR>\n" );
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document.write( "in this site.\r<BR>\n" );
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document.write( "Also, you have this free of charge online textbook in ALGEBRA-II in this site\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>.\r<BR>\n" );
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document.write( "The referred lesson is the part of this textbook under the topic<BR>\n" );
document.write( "\"<U>Binomial expansion, binomial coefficients, Pascal's triangle</U>\".\r<BR>\n" );
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document.write( "Save the link to this textbook together with its description\r<BR>\n" );
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document.write( "Free of charge online textbook in ALGEBRA-II<BR>\n" );
document.write( "https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson\r<BR>\n" );
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document.write( "into your archive and use when it is needed.\r<BR>\n" );
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