document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1019009>Question 1019009</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a committee of 5 people must be selected from 5 men and 8 women. 13 people in ALL.\r<BR>\n" );
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document.write( "How many ways can a selection be done if there are at least 3 women on the committee? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #634988 by </B><A HREF=/tutors/aboutme.mpl?userid=mathmate><B>mathmate(429)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathmate><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=mathmate><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=634988\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Question:<BR>\n" );
document.write( "a committee of 5 people must be selected from 5 men and 8 women. 13 people in ALL.<BR>\n" );
document.write( "How many ways can a selection be done if there are at least 3 women on the committee?<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "There are three cases:<BR>\n" );
document.write( "1. 3 women and two men <BR>\n" );
document.write( "C(5,2)*C(8,3)=560 ways<BR>\n" );
document.write( "2. 4 women and 1 man:<BR>\n" );
document.write( "C(5,1)*C(8,4)=350 ways<BR>\n" );
document.write( "3. 5 women and no men:<BR>\n" );
document.write( "C(5,0)*C(8,5)=56 ways<BR>\n" );
document.write( "Total=560+350+56=<b>966</b> ways.<BR>\n" );
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document.write( "Following is the <i>incorrect</i> way to solve the problem:<BR>\n" );
document.write( "Choose 3 women and two more from the rest:<BR>\n" );
document.write( "C(8,3)*C(10,2)=2520 >>966<BR>\n" );
document.write( "This way, we're overcounting, since some of the women chosen out of the remaining 10 could have been counted already, for example, ABC were chosen first, and G was chosen next.  But then it would be the same if ABG was chosen first, and C was chosen later.<BR>\n" );
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