document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=573882>Question 573882</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 6 friends go out for dinner and order from a menu of 8 different sandwiches. What is the probability that at least 2 of the group order the same sandwich? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #368906 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=368906\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "There are exactly two possibilities:  Either at least 2 have the same sandwich or everyone has a different sandwich.  Hence the probability that at least 2 have the same sandwich is equal to 1 minus the probability that they all have different sandwiches.  So what, you say?  Calculating the probability that they all have different sandwiches is a much simpler calculation, that's so what.\r<BR>\n" );
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document.write( "The probability that the first guy will order a different sandwich than everyone else is certainty.  There are eight different sandwiches and he can choose any of the eight without duplicating anyone.\r<BR>\n" );
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document.write( "The second guy only has 7 sandwiches to choose because one of the possible 8 was already used by the first guy, hence his probability of not duplicating is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \frac{7}{8}\">\r<BR>\n" );
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document.write( "Repeating the pattern through all six guys, the probability of no duplication is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{8}{8}\ \times\ \frac{7}{8}\ \times\ \frac{6}{8}\ \times\ \frac{5}{8}\ \times\ \frac{4}{8}\ \times\ \frac{3}{8}\ =\ \frac{8!}{(8\,-\,6)!8^6}\">\r<BR>\n" );
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document.write( "So the probability of at least one duplication is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 1\ -\ \frac{8!}{(8\,-\,6)!8^6}\">\r<BR>\n" );
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document.write( "You can do the arithmetic.\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" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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