document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=313187>Question 313187</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Five lotto numbers are to be picked at random from a collection of 100 numbers.  Order does not matter.  How many combinations of lotto numbers are there? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #223946 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=223946\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "You need the number of ways to select <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large k\"> things from a selection of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large n\"> things where order does not matter.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(n\cr k\right\)\ =\ \frac{n!}{k!(n\,-\,k)!}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \left(100\cr\ \,5\right\)\ =\ \frac{100!}{5!(100\,-\,5)!}\">\r<BR>\n" );
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document.write( "Which is some very nasty arithmetic.  However, there is a shortcut...\r<BR>\n" );
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document.write( "If you divide <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 100!\"> by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 95!\"> you get <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 100\ \times\ 99\ \times\ 98\ \times\ 97\ \times\ 96\">\r<BR>\n" );
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document.write( "Then divide this product by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 5!\ =\ 5\ \times\ 4\ \times\ 3\ \times\ 2\ \times\ 1\">\r<BR>\n" );
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document.write( "The 5 goes into 100 20 times, the 4 goes into 96 24 times, the 3 goes into 99 33 times and the 2 goes into 98 49 times, so the whole thing reduces to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 20\ \times\ 33\ \times\ 49\ \times\ 97\ \times\ 24\">\r<BR>\n" );
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document.write( "Somewhat simpler.  You can take it from here.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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