document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205092>Question 1205092</A>: <I><SPAN STYLE=\"word-break: break-all;\"> #1. <BR>\n" );
document.write( "Prove for all integers n, k, and r with n ≥ k ≥ r that nCk×kCr = nCr×(n-r)C(k-r)\r<BR>\n" );
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document.write( "#2. <BR>\n" );
document.write( "The binomial theorem states that for any real numbers a and b,<BR>\n" );
document.write( "(a + b)n =∑_(k=0)^n▒〖(n¦k) a^(n-k) b^k 〗 for any integer n ≥ 0.<BR>\n" );
document.write( "Use this theorem to show that for any integer n ≥ 0, ∑_(k=0)^n▒〖〖(-1)〗^k (n¦k) 3^(n-k) 2^k 〗 = 1.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #841735 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=841735\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Tutor mccravyedwin has covered problem 1. <BR>\n" );
document.write( "I'll take a look at problem 2.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%2Bb%29%5En+=+sum%28nCk%2Aa%5E%28n-k%29%2Ab%5Ek%2Ck=0%2Cn%29 ALIGN=MIDDLE  ALT=\"%28a%2Bb%29%5En+=+sum%28nCk%2Aa%5E%28n-k%29%2Ab%5Ek%2Ck=0%2Cn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > The binomial theorem where n is an integer and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+%3E=+0 ALIGN=MIDDLE  ALT=\"n+%3E=+0\"   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=%283%2B%28-2%29%29%5En+=+sum%28nCk%2A3%5E%28n-k%29%2A%28-2%29%5Ek%2Ck=0%2Cn%29 ALIGN=MIDDLE  ALT=\"%283%2B%28-2%29%29%5En+=+sum%28nCk%2A3%5E%28n-k%29%2A%28-2%29%5Ek%2Ck=0%2Cn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in a = 3 and b = -2\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283-2%29%5En+=+sum%28nCk%2A3%5E%28n-k%29%2A%28-1%2A2%29%5Ek%2Ck=0%2Cn%29 ALIGN=MIDDLE  ALT=\"%283-2%29%5En+=+sum%28nCk%2A3%5E%28n-k%29%2A%28-1%2A2%29%5Ek%2Ck=0%2Cn%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=%281%29%5En+=+sum%28nCk%2A3%5E%28n-k%29%2A%28-1%29%5Ek%2A2%5Ek%2Ck=0%2Cn%29 ALIGN=MIDDLE  ALT=\"%281%29%5En+=+sum%28nCk%2A3%5E%28n-k%29%2A%28-1%29%5Ek%2A2%5Ek%2Ck=0%2Cn%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=1+=+sum%28%28-1%29%5Ek%2AnCk%2A3%5E%28n-k%29%2A2%5Ek%2Ck=0%2Cn%29 ALIGN=MIDDLE  ALT=\"1+=+sum%28%28-1%29%5Ek%2AnCk%2A3%5E%28n-k%29%2A2%5Ek%2Ck=0%2Cn%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=sum%28%28-1%29%5Ek%2AnCk%2A3%5E%28n-k%29%2A2%5Ek%2Ck=0%2Cn%29+=+1 ALIGN=MIDDLE  ALT=\"sum%28%28-1%29%5Ek%2AnCk%2A3%5E%28n-k%29%2A2%5Ek%2Ck=0%2Cn%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+%3E=+0 ALIGN=MIDDLE  ALT=\"n+%3E=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The nCk refers to the combination formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=nCk+=+%28n%21%29%2F%28k%21%2A%28n-k%29%21%29 ALIGN=MIDDLE  ALT=\"nCk+=+%28n%21%29%2F%28k%21%2A%28n-k%29%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The nCk values are found in Pascal's Triangle.<BR>\n" );
document.write( "The nCk replaces the notation (n¦k) which is often written as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2Cn%2Ck%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2Cn%2Ck%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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