document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200910>Question 1200910</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A club with 20 women and 17 men needs to form a committee of size six.<BR>\n" );
document.write( "How many committees are possible?<BR>\n" );
document.write( "How many committees are possible if the committee must have three women and three men?<BR>\n" );
document.write( "How many committees are possible if the committee must have at least two men?<BR>\n" );
document.write( "How many committees are possible if the committee must consist of all women or all men?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #835142 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=835142\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
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document.write( "Question 1<BR>\n" );
document.write( "A club with 20 women and 17 men needs to form a committee of size six.<BR>\n" );
document.write( "How many committees are possible?<BR>\n" );
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document.write( "The seats on the committee aren't named and have equal rank. <BR>\n" );
document.write( "This means the order doesn't matter.<BR>\n" );
document.write( "We use the nCr combination formula.\r<BR>\n" );
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document.write( "n = 20 women + 17 men = 37 people total<BR>\n" );
document.write( "r = 6 selections<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "37 C 6 = (37!)/(6!*(37-6)!)<BR>\n" );
document.write( "37 C 6 = (37!)/(6!*31!)<BR>\n" );
document.write( "37 C 6 = (37*36*35*34*33*32*31!)/(6!*31!)<BR>\n" );
document.write( "37 C 6 = (37*36*35*34*33*32)/(6!)<BR>\n" );
document.write( "37 C 6 = (37*36*35*34*33*32)/(6*5*4*3*2*1)<BR>\n" );
document.write( "37 C 6 = (1673844480)/(720)<BR>\n" );
document.write( "37 C 6 = 2324784\r<BR>\n" );
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document.write( "Answer: <font color=red>2,324,784</font><BR>\n" );
document.write( "This is roughly 2.3 million\r<BR>\n" );
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document.write( "Question 2<BR>\n" );
document.write( "A club with 20 women and 17 men needs to form a committee of size six.<BR>\n" );
document.write( "How many committees are possible if the committee must have three women and three men?<BR>\n" );
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document.write( "There are n = 20 women and there are r = 3 selections.<BR>\n" );
document.write( "Use the nCr combination formula.<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "20 C 3 = (20!)/(3!*(20-3)!)<BR>\n" );
document.write( "20 C 3 = (20!)/(3!*17!)<BR>\n" );
document.write( "20 C 3 = (20*19*18*17!)/(3!*17!)<BR>\n" );
document.write( "20 C 3 = (20*19*18)/(3!)<BR>\n" );
document.write( "20 C 3 = (20*19*18)/(3*2*1)<BR>\n" );
document.write( "20 C 3 = (6840)/(6)<BR>\n" );
document.write( "20 C 3 = 1140<BR>\n" );
document.write( "Take note of the 20*19*18 in the numerator and 3! = 3*2*1 = 6 in the denominator.<BR>\n" );
document.write( "There are 1140 ways to pick the three women where order doesn't matter.\r<BR>\n" );
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document.write( "Follow similar steps for the men.<BR>\n" );
document.write( "n = 17<BR>\n" );
document.write( "r = 3<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "17 C 3 = (17!)/(3!*(17-3)!)<BR>\n" );
document.write( "17 C 3 = (17!)/(3!*14!)<BR>\n" );
document.write( "17 C 3 = (17*16*15*14!)/(3!*14!)<BR>\n" );
document.write( "17 C 3 = (17*16*15)/(3!)<BR>\n" );
document.write( "17 C 3 = (17*16*15)/(3*2*1)<BR>\n" );
document.write( "17 C 3 = (4080)/(6)<BR>\n" );
document.write( "17 C 3 = 680<BR>\n" );
document.write( "There are 680 ways to pick the three men where order doesn't matter.\r<BR>\n" );
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document.write( "Summary:<BR>\n" );
document.write( "1140 ways to pick the three women<BR>\n" );
document.write( "680 ways to pick the three men\r<BR>\n" );
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document.write( "Therefore, 1140*680 = 775,200 is the number of ways to have a committee of 3 women and 3 men. Order doesn't matter.\r<BR>\n" );
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document.write( "Answer: <font color=red>775,200</font>\r<BR>\n" );
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document.write( "Question 3<BR>\n" );
document.write( "A club with 20 women and 17 men needs to form a committee of size six.<BR>\n" );
document.write( "How many committees are possible if the committee must have at least two men?<BR>\n" );
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document.write( "Let's consider the event \"at most 1 man\".<BR>\n" );
document.write( "It can be broken down into these two cases:<BR>\n" );
document.write( "Case A: 6 women and 0 men on the committee<BR>\n" );
document.write( "Case B: 5 women and 1 man on the committee\r<BR>\n" );
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document.write( "Case A:<BR>\n" );
document.write( "20C6 = 38,760 ways to pick the women<BR>\n" );
document.write( "17C0 = 1 way to pick the men (i.e. 1 way to leave the men out entirely)<BR>\n" );
document.write( "38,760*1 = 38,760 different ways to do case A.\r<BR>\n" );
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document.write( "Case B:<BR>\n" );
document.write( "20C5 = 15,504 ways to pick the women<BR>\n" );
document.write( "17C1 = 17 ways to pick the men<BR>\n" );
document.write( "15,504*17 = 263,568 different ways to do case B.\r<BR>\n" );
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document.write( "We have<BR>\n" );
document.write( "A+B = 38,760+263,568 = 302,328<BR>\n" );
document.write( "ways to have at most 1 man.\r<BR>\n" );
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document.write( "This is out of 37C6 = 2,324,784 ways to form the committee (refer to question 1).\r<BR>\n" );
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document.write( "2,324,784 - 302,328 = 2,022,456<BR>\n" );
document.write( "This is the number of ways to have at least two men. \r<BR>\n" );
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document.write( "The events \"at most 1 man\" and \"at least two men\" are complementary. <BR>\n" );
document.write( "They make up all possible outcomes.\r<BR>\n" );
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document.write( "Answer: <font color=red>2,022,456</font>\r<BR>\n" );
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document.write( "Question 4<BR>\n" );
document.write( "A club with 20 women and 17 men needs to form a committee of size six.<BR>\n" );
document.write( "How many committees are possible if the committee must consist of all women or all men?<BR>\n" );
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document.write( "Let's determine how many committees are possible if the committee consists of women only.<BR>\n" );
document.write( "n = 20 women<BR>\n" );
document.write( "r = 6 selections<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "20 C 6 = (20!)/(6!*(20-6)!)<BR>\n" );
document.write( "20 C 6 = (20!)/(6!*14!)<BR>\n" );
document.write( "20 C 6 = (20*19*18*17*16*15*14!)/(6!*14!)<BR>\n" );
document.write( "20 C 6 = (20*19*18*17*16*15)/(6!)<BR>\n" );
document.write( "20 C 6 = (20*19*18*17*16*15)/(6*5*4*3*2*1)<BR>\n" );
document.write( "20 C 6 = (27907200)/(720)<BR>\n" );
document.write( "20 C 6 = 38760\r<BR>\n" );
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document.write( "Now let's see how many committees are possible if the committee consists of men only.<BR>\n" );
document.write( "n = 17 men<BR>\n" );
document.write( "r = 6 selections<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "17 C 6 = (17!)/(6!*(17-6)!)<BR>\n" );
document.write( "17 C 6 = (17!)/(6!*11!)<BR>\n" );
document.write( "17 C 6 = (17*16*15*14*13*12*11!)/(6!*11!)<BR>\n" );
document.write( "17 C 6 = (17*16*15*14*13*12)/(6!)<BR>\n" );
document.write( "17 C 6 = (17*16*15*14*13*12)/(6*5*4*3*2*1)<BR>\n" );
document.write( "17 C 6 = (8910720)/(720)<BR>\n" );
document.write( "17 C 6 = 12376\r<BR>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "<font color=red>38,760</font> ways to have a committee of all women.<BR>\n" );
document.write( "<font color=red>12,376</font> ways to have a committee of all men.<BR>\n" );
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