document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1038081>Question 1038081</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Show that nC1-2nC2+3nC3+...(-1)n+1 nCn=0. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #652810 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=652810\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Consider the binomial expansion  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x-1%29%5En+=+sum%28%28matrix%282%2C1%2Cn%2Ck%29%29%2Ax%5E%28n-k%29%2A%28-1%29%5Ek%2C+k=0%2Cn%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Taking the derivatives of both sides of the preceding equation, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%28x-1%29%5E%28n-1%29+=+sum%28%28matrix%282%2C1%2Cn%2Ck%29%29%2A%28n-k%29%2Ax%5E%28n-k-1%29%2A%28-1%29%5Ek%2C+k=0%2Cn-1%29+=sum%28%28+n%21%2F%28%28n-k-1%29%21k%21%29%29%2Ax%5E%28n-k-1%29%2A%28-1%29%5Ek%2C+k=0%2Cn-1%29+=++sum%28%28+%28%28k%2B1%29n%21%29%2F%28%28n-k-1%29%21%28k%2B1%29%21%29%29%2Ax%5E%28n-k-1%29%2A%28-1%29%5Ek%2C+k=0%2Cn-1%29+=+sum%28%28k%2B1%29%2A%28matrix%282%2C1%2Cn%2Ck%2B1%29%29%2Ax%5E%28n-k-1%29%2A%28-1%29%5Ek%2C+k=0%2Cn-1%29++ ALIGN=MIDDLE   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=sum%28k%2A%28matrix%282%2C1%2Cn%2Ck%29%29%2Ax%5E%28n-k-2%29%2A%28-1%29%5E%28k-1%29%2C+k=1%2Cn%29++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now let x = 1:\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+=+n%281-1%29%5E%28n-1%29+=+sum%28k%2A%28matrix%282%2C1%2Cn%2Ck%29%29%2A%28-1%29%5E%28k-1%29%2C+k=1%2Cn%29+=++sum%28k%2A%28matrix%282%2C1%2Cn%2Ck%29%29%2A%28-1%29%5E%28k%2B1%29%2C+k=1%2Cn%29++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >,  \r<BR>\n" );
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document.write( "since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%29%5E%28k-1%29+=+%28-1%29%5E%28k%2B1%29 ALIGN=MIDDLE  ALT=\"%28-1%29%5E%28k-1%29+=+%28-1%29%5E%28k%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "And the identity is proved...\r<BR>\n" );
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document.write( " </SPAN>\n" );
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