document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176410>Question 1176410</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Question: There are 5 physicists, 4 chemists, and 3 mathematicians. In how many ways can the committee of 6 be chosen so that AT LEAST 3 members of the committee are physicists? <BR>\n" );
document.write( "My answer: C(5,3) =  10. <BR>\n" );
document.write( "Once the three physicists are selected, since it is AT LEAST, the remaining 2 physicists can still be selected which makes available candidates for the remaining 3 positions n = 9<BR>\n" );
document.write( "We need another 3. so C(9,3) = 84.<BR>\n" );
document.write( "Therefore, the number of ways the committee could be chosen is 10 X 84 = 840<BR>\n" );
document.write( "But the book says 462.<BR>\n" );
document.write( "What am I doing wrong?<BR>\n" );
document.write( "Thanks for your help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #803222 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=803222\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 12C6 are the number of possible ways to choose 6 <BR>\n" );
document.write( "at least 3 physicists would be<BR>\n" );
document.write( "3: 5C3*7C3 for the other 3 which is 10*35 or 350. Consider once you have 5C3 in the numerator, you need another 7 on top and 3 on the bottom.<BR>\n" );
document.write( "4, 5C4*7C2 which would be 105. Notice how the two change<BR>\n" );
document.write( "5 5C5*7C1, or 7<BR>\n" );
document.write( "add to 462.<BR>\n" );
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document.write( "The candidates for the other 3 positions can't be physicists so the choices are 7C3.<BR>\n" );
document.write( "And at least 3 members means 4 and 5 can be physicists as well.<BR>\n" );
document.write( "You are assuming the first 3 are physicists, and yes, that leaves 9. But if only 3 are required, then it leaves 7, not 9 people, since a 4th physicist won't be allowed.<BR>\n" );
document.write( "If 4 are required, it leaves 7 as well. <BR>\n" );
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