document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=477451>Question 477451</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dear math teacher, \r<BR>\n" );
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document.write( "I am having difficulties solving the following problem (#92b)in the fastest and least time consuming way.  I got the result but when I tried the second method, my answers did not match.  Would you please explain why?  \r<BR>\n" );
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document.write( "Here is the problem:\r<BR>\n" );
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document.write( "\"91. From 5 physicists, 4 chemists and 3 mathematicians a committee of 6 is to be chosen so as to include 3 physicists, 2 chemists and 1 mathematician.  In how many ways can this be done?\"\r<BR>\n" );
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document.write( "\"92. In problem 91, in how many ways can the committee of 6 be chosen so that<BR>\n" );
document.write( "a) 2 members of the committee are mathematicians, <BR>\n" );
document.write( "b) at least 3 members of the committee are physicists? \r<BR>\n" );
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document.write( "   P    Chem   Math<BR>\n" );
document.write( "n  5    4     3<BR>\n" );
document.write( "r  3<BR>\n" );
document.write( "   4<BR>\n" );
document.write( "   5<BR>\n" );
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document.write( "   3    3     3  thus, 5C3 times 4C3 times 3C0 = 10.4.1 = 40<BR>\n" );
document.write( "   3    0     3        5C3.4C0.3C3 = 10.1.1 = 10<BR>\n" );
document.write( "   3    1     2        5C3.4C1.3C2 = 10.4.3 = 120<BR>\n" );
document.write( "   3    2     1        5C3.4C2.3C1 = 10.6.3 = 180<BR>\n" );
document.write( "                                      Total = 350\r<BR>\n" );
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document.write( "   4    2     0        5C4.4C2.3C0 = 30<BR>\n" );
document.write( "   4    0     2        5C4.4C0.3C2 = 15<BR>\n" );
document.write( "   4    1     1        5C4.4C1.3C1 = 60<BR>\n" );
document.write( "                             Total = 105\r<BR>\n" );
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document.write( "   5   1      0        5C5.4C1.3C0 = 1.4.1 = 4<BR>\n" );
document.write( "   5   0      1        5C5.4C0.3C1 = 1.1.3 = 3<BR>\n" );
document.write( "                                     Total = 7\r<BR>\n" );
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document.write( "350 + 105 + 7 = 462\r<BR>\n" );
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document.write( "_______<BR>\n" );
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document.write( "Now method 2:\r<BR>\n" );
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document.write( "Total # of ways to select committee of 6 out of 12 = <BR>\n" );
document.write( "12C6 = 924 ways\r<BR>\n" );
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document.write( "Non-physicists can be selected in 12-5C6 ways = 7!/(7-6)!6! = 7 ways\r<BR>\n" );
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document.write( "I am not really sure how to continue solving this using this method.  How can you correct me?  Please let me know. \r<BR>\n" );
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document.write( "Thank you very much. \r<BR>\n" );
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document.write( "Yours sincerely, \r<BR>\n" );
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document.write( "Ivanka   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #327443 by </B><A HREF=/tutors/aboutme.mpl?userid=sudhanshu_kmr><B>sudhanshu_kmr(1152)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=sudhanshu_kmr><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=sudhanshu_kmr><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=327443\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "b) at least 3 members of the committee are physicists. \r<BR>\n" );
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document.write( "possible ways are : \r<BR>\n" );
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document.write( "1. 3 physicists and 3 from chemists and mathematicians\r<BR>\n" );
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document.write( "2. 4 physicists and 2 from chemists and mathematicians\r<BR>\n" );
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document.write( "3. 5 physicists and 1 from chemists and mathematicians\r<BR>\n" );
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document.write( "total number of ways = 5C3 * 7C3 + 5C4 * 7C2 + 5C5 * 7C1 \r<BR>\n" );
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document.write( "if any doubt, you are welcome to contact...<BR>\n" );
document.write( "  </SPAN>\n" );
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