document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1169982>Question 1169982</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You volunteer to help drive children at a charity event to the​ zoo, but you can fit only 6 of the 15 children present in your van. How many different groups of  children can you​ drive?<BR>\n" );
document.write( "How many different groups of 6 children can you​ drive?<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 #794844 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=794844\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Consider 6 slots labeled A,B,C,D,E,F to represent the 6 people you select from the 15 total.\r<BR>\n" );
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document.write( "We have...<BR>\n" );
document.write( "15 choices for slot A<BR>\n" );
document.write( "14 choices for slot B<BR>\n" );
document.write( "13 choices for slot C<BR>\n" );
document.write( "12 choices for slot D<BR>\n" );
document.write( "11 choices for slot E<BR>\n" );
document.write( "10 choices for slot F<BR>\n" );
document.write( "Note how I'm counting down one at a time. This is because once a student is selected, they can't be picked again. \r<BR>\n" );
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document.write( "Multiplying out those items gives us<BR>\n" );
document.write( "15*14*13*12*11*10 = 3,603,600<BR>\n" );
document.write( "which is the number of permutations. \r<BR>\n" );
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document.write( "If order mattered, then we'd stop here. However, order does not matter. Each seat is unlabeled and there is no ranking. All that matters is the overall group.\r<BR>\n" );
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document.write( "For any group of 6 people, there are 6! = 6*5*4*3*2*1 = 720 ways to arrange said group. We must divide the permutation count found above to get the answer we want.\r<BR>\n" );
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document.write( "number of combinations = (number of permutations)/720<BR>\n" );
document.write( "number of combinations = (3,603,600)/720<BR>\n" );
document.write( "number of combinations = 5,005\r<BR>\n" );
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document.write( "We could also use the nCr combination formula\r<BR>\n" );
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document.write( "We have a pool size of n = 15 and r = 6 selections to make from that pool size.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large _n C_r = \frac{n!}{r!*(n-r)!}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large _{15} C_6 = \frac{15!}{6!*(15-6)!}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large _{15} C_6 = \frac{15!}{6!*9!}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large _{15} C_6 = \frac{15*14*13*12*11*10*9!}{6!*9!}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large _{15} C_6 = \frac{15*14*13*12*11*10}{6!}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large _{15} C_6 = \frac{15*14*13*12*11*10}{6*5*4*3*2*1}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large _{15} C_6 = \frac{3,603,600}{720}\"> This expression of (3,603,600)/720 shows up again\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large _{15} C_6 = 5,005\">\r<BR>\n" );
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document.write( "Answer: 5005<BR>\n" );
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