document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=957780>Question 957780</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a committee of 5 people is to be chosen from 10 men and 12 women. How many ways can the committee be formed if it is to contain at least 2 women, AND one particular man (M1) and one particular woman(W1) DO NOT want to serve together?\r<BR>\n" );
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document.write( "I know at least two women will be (12C2)(20C3) and I think I need to subtract the number of committees that include both W1 and M1 from this number, OR add up the committees consisting of <BR>\n" );
document.write( "(M1 alone) + (W1 alone) + (neither M1 or W1)  but I'm stumped on how to proceed </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #585310 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=585310\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> looks like you have 2 problems here.\r<BR>\n" );
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document.write( "the first problem is you need to include at least 2 women.<BR>\n" );
document.write( "this means you can include 2 women or 3 women or 4 women or 5 women on the committee since there is no requirement on the number of males that have to be on the committee.\r<BR>\n" );
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document.write( "the second problem is that you cannot have a particular woman paired with a particular man.\r<BR>\n" );
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document.write( "we'll tackle the first problem first and then we'll tackle the second problem.\r<BR>\n" );
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document.write( "there are 10 men and 12 women.\r<BR>\n" );
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document.write( "if 3 men and 2 women are on the committee, then the possible number of combinations is c((10,3) * c(12,2) which is equal to 7920.\r<BR>\n" );
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document.write( "if 2 men and 3 women are on the committee, then the possible number of combinations is c(10,2) * c(12,3) which is equal to 9900.\r<BR>\n" );
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document.write( "if 1 man and 4 women are on the committee, then the possible number of combinations is c(10,1) * c(12,4) which is equal to 4950.\r<BR>\n" );
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document.write( "if no men and 5 women are on the committee, then the possible number of combinations is c(10,0) * c(12,5) which is equal to 792.\r<BR>\n" );
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document.write( "you would need to add these up to get the total number of possible combinations.\r<BR>\n" );
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document.write( "7920 + 9900 + 4950 + 792 = 25362\r<BR>\n" );
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document.write( "now we'll tackle the second problem.\r<BR>\n" );
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document.write( "a particular women does not want to be with a particular man.\r<BR>\n" );
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document.write( "you would need to determine how many possible combinations this can happen in and subtract them from the total number of possible combinations.\r<BR>\n" );
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document.write( "if there are 2 women and 3 men on the committee, then this particular woman can be paired with 1 out of the 11 remaining women and this particular man can be paired with 2 of the 9 remaining men.\r<BR>\n" );
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document.write( "the total number of ways this can happen is c(11,1) * c(9,2) = 396.\r<BR>\n" );
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document.write( "with 3 women and 2 men, the total number of ways this can happen is c(11,2) * c(9,1) = 495.\r<BR>\n" );
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document.write( "with 4 women and 1 man, the total number of ways this can happen is c(11,3) * c(9,0) = 165.\r<BR>\n" );
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document.write( "with 5 women and 0 men, the total number of ways this can happen is c(11,4) * 0 = 0.\r<BR>\n" );
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document.write( "you need to add all these up and then subtract them from the total number of possible combinations to get your answer.\r<BR>\n" );
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document.write( "396 + 495 + 165 + 0 = 1056.\r<BR>\n" );
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document.write( "25362 - 1056 = 24306.\r<BR>\n" );
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document.write( "there are 24306 ways at least 2 women can be combined with men to form a committee of 5 people where one particular woman is not paired with one particular man.\r<BR>\n" );
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document.write( "that's my understanding of how this needs to be solved.\r<BR>\n" );
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document.write( "i worked it out with much smaller numbers and the method appears good.\r<BR>\n" );
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document.write( "c(n,x) is the combination formula of n! / (x! * (n-x)!)\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "c(9,2) is equal to 9! / (2! * 7!) which is equal to 36.<BR>\n" );
document.write( "c(11,1) is equal to 11! / (1! * 10!) which is equal to 11.<BR>\n" );
document.write( "11 * 36 = 396.\r<BR>\n" );
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document.write( "you saw this in the following statement copied from above:\r<BR>\n" );
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document.write( "the total number of ways this can happen is c(11,1) * c(9,2) = 396.\r<BR>\n" );
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