document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=970351>Question 970351</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that C(n,r)= C(n-1,r-1)+C(n-1,r)\r<BR>\n" );
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document.write( "NOTE: Professor said that you can start in either left side or right side </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #592999 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=592999\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2Cr%29+=+%28n%21%29%2F%28r%21%2A%28n-r%29%21%29 ALIGN=MIDDLE  ALT=\"C%28n%2Cr%29+=+%28n%21%29%2F%28r%21%2A%28n-r%29%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > by definition\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2Cr%29+=+%28n%21%29%2F%28r%21%2A%28n-r%29%21%29 ALIGN=MIDDLE  ALT=\"C%28n%2Cr%29+=+%28n%21%29%2F%28r%21%2A%28n-r%29%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n-1%2Cr-1%29+=+%28%28n-1%29%21%29%2F%28%28r-1%29%21%2A%28+%28n-1%29+-+%28r-1%29+%29%21%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Replace every 'n' with 'n-1'. Replace every 'r' with 'r-1'\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n-1%2Cr-1%29+=+%28%28n-1%29%21%29%2F%28%28r-1%29%21%2A%28+n+-+1+-+r+%2B+1+%29%21%29 ALIGN=MIDDLE  ALT=\"C%28n-1%2Cr-1%29+=+%28%28n-1%29%21%29%2F%28%28r-1%29%21%2A%28+n+-+1+-+r+%2B+1+%29%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n-1%2Cr-1%29+=+%28%28n-1%29%21%29%2F%28%28r-1%29%21%2A%28n+-+r%29%21%29 ALIGN=MIDDLE  ALT=\"C%28n-1%2Cr-1%29+=+%28%28n-1%29%21%29%2F%28%28r-1%29%21%2A%28n+-+r%29%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > I'm going to call this equation Q\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n-1%2Cr%29+=+%28%28n-1%29%21%29%2F%28r%21%2A%28+n+-+1+-+r+%29%21%29 ALIGN=MIDDLE  ALT=\"C%28n-1%2Cr%29+=+%28%28n-1%29%21%29%2F%28r%21%2A%28+n+-+1+-+r+%29%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace every 'n' with 'n-1'. I'm going to call this equation R\r<BR>\n" );
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document.write( "we will use these ideas\r<BR>\n" );
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document.write( "to help us do the proof.\r<BR>\n" );
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document.write( "Now onto the main proof. I'm only going to manipulate the right side of the equation to transform it into the left side.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2Cr%29+=+%28%28n-1%29%21%29%2F%28%28r-1%29%21%2A%28n+-+r%29%21%29+%2B+%28%28n-1%29%21%29%2F%28r%21%2A%28+n+-+1+-+r+%29%21%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Substitute equation Q and equation R (see above)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2Cr%29+=+%28%28n-1%29%21%29%2F%28%28r-1%29%21%2A%28n+-+r%29%28n+-+r+-+1%29%21%29+%2B+%28%28n-1%29%21%29%2F%28r%2A%28r-1%29%21%2A%28+n++-+r+-+1+%29%21%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Use the tricks shown above (eg: to write <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%21 ALIGN=MIDDLE  ALT=\"r%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%2A%28r-1%29%21 ALIGN=MIDDLE  ALT=\"r%2A%28r-1%29%21\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2Cr%29+=+%28red%28%28n-1%29%21%29%29%2F%28red%28%28r-1%29%21%29%2A%28n+-+r%29red%28%28n-r-1%29%21%29%29+%2B+%28red%28%28n-1%29%21%29%29%2F%28r%2Ared%28%28r-1%29%21%29%2Ared%28%28n-r-1%29%21%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > I'm highlighting the common terms (shared between the fractions) in <font color=\"red\">red</font>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2Cr%29+=+%28%28+%28n-1%29%21+%29%2F%28+%28r-1%29%21+%2A+%28n+-+r+-+1%29%21+%29%29%2A%28+1%2F%28n-r%29+%2B+1%2Fr+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Factor out the common terms\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2Cr%29+=+C%28n%2Cr%29 ALIGN=MIDDLE  ALT=\"C%28n%2Cr%29+=+C%28n%2Cr%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > So the identity is confirmed. </SPAN>\n" );
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