document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196992>Question 1196992</A>: <I><SPAN STYLE=\"word-break: break-all;\"> All the members of a group of 30 students belong to at least one of these clubs: Academic, Music and Arts.  <BR>\n" );
document.write( " 6 of the students belong to only the Arts Club.  <BR>\n" );
document.write( " 5 of the students belong to all 3 clubs.  <BR>\n" );
document.write( "      2 of the students belong to the Academic and Arts clubs but not to the Music club.  <BR>\n" );
document.write( "      15 of the students belong to the Arts Club.  <BR>\n" );
document.write( "      2of the students belong only to the Academic Club.  <BR>\n" );
document.write( "      3 of the students belong only to the Music Club.  <BR>\n" );
document.write( "Find the number of students who belong to at least two clubs?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #830048 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=830048\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "A = arts club<BR>\n" );
document.write( "B = academic club<BR>\n" );
document.write( "C = music club\r<BR>\n" );
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document.write( "This is what the Venn Diagram looks like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/9jI1r86.png\"><BR>\n" );
document.write( "We have these 8 distinct regions marked in red<BR>\n" );
document.write( "a = number in arts club only<BR>\n" );
document.write( "b = number in arts & academics, but not music<BR>\n" );
document.write( "c = number in academics club only<BR>\n" );
document.write( "d = number in arts & music, but not academic<BR>\n" );
document.write( "e = number of students in all 3 clubs<BR>\n" );
document.write( "f = number in academics & music, but not arts<BR>\n" );
document.write( "g = number in music club only<BR>\n" );
document.write( "h = number of students in neither of the 3 clubs mentioned\r<BR>\n" );
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document.write( "I'll break the given info into these facts<BR>\n" );
document.write( "<font color=blue>Fact [1]</font>: 6 of the students belong to only the Arts Club.<BR>\n" );
document.write( "<font color=blue>Fact [2]</font>: 5 of the students belong to all 3 clubs.<BR>\n" );
document.write( "<font color=blue>Fact [3]</font>: 2 of the students belong to the Academic and Arts clubs but not to the Music club.<BR>\n" );
document.write( "<font color=blue>Fact [4]</font>: 15 of the students belong to the Arts Club.<BR>\n" );
document.write( "<font color=blue>Fact [5]</font>: 2of the students belong only to the Academic Club.<BR>\n" );
document.write( "<font color=blue>Fact [6]</font>: 3 of the students belong only to the Music Club.\r<BR>\n" );
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document.write( "Based on those facts we can then determine the following:<BR>\n" );
document.write( "a = 6 .... from <font color=blue>Fact [1]</font><BR>\n" );
document.write( "b = 2 .... from <font color=blue>Fact [3]</font><BR>\n" );
document.write( "c = 2 .... from <font color=blue>Fact [5]</font><BR>\n" );
document.write( "d = unknown for now<BR>\n" );
document.write( "e = 5 .... from <font color=blue>Fact [2]</font><BR>\n" );
document.write( "f = unknown for now<BR>\n" );
document.write( "g = 3 .... from <font color=blue>Fact [6]</font><BR>\n" );
document.write( "h = 0 ... because every student mentioned belongs to at least one of the three clubs\r<BR>\n" );
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document.write( "We've used nearly every fact, except fact 4, to determine values a,b,c,e,g,and h.\r<BR>\n" );
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document.write( "Use <font color=blue>Fact [4]</font> to form and solve this equation<BR>\n" );
document.write( "a+b+d+e = 15 students in the arts club (circle A)<BR>\n" );
document.write( "6+2+d+5 = 15<BR>\n" );
document.write( "13+d = 15<BR>\n" );
document.write( "d = 15-13<BR>\n" );
document.write( "d = 2\r<BR>\n" );
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document.write( "Now we'll use the fact that there are 30 students total, each of which belongs to one or more clubs.<BR>\n" );
document.write( "a+b+c+d+e+f+g+h = 30<BR>\n" );
document.write( "6+2+2+2+5+f+3+0 = 30<BR>\n" );
document.write( "20+f = 30<BR>\n" );
document.write( "f = 30-20<BR>\n" );
document.write( "f = 10\r<BR>\n" );
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document.write( "Here's a summary of all the values<BR>\n" );
document.write( "a = 6<BR>\n" );
document.write( "b = 2<BR>\n" );
document.write( "c = 2<BR>\n" );
document.write( "d = 2<BR>\n" );
document.write( "e = 5<BR>\n" );
document.write( "f = 10<BR>\n" );
document.write( "g = 3<BR>\n" );
document.write( "h = 0<BR>\n" );
document.write( "and here's what the Venn diagram looks like now after replacing the letters with their corresponding numbers.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/K9TB1HE.png\">\r<BR>\n" );
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document.write( "We're now told to \"Find the number of students who belong to at least two clubs\".\r<BR>\n" );
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document.write( "There are two ways to do this:\r<BR>\n" );
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document.write( "Method 1)\r<BR>\n" );
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document.write( "Add up the values b, d, e, f as they represent students who are in 2 clubs or more.<BR>\n" );
document.write( "b+d+e+f = 2+2+5+10 = 19<BR>\n" );
document.write( "There are 19 students who are in 2 clubs or more.\r<BR>\n" );
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document.write( "Method 2)\r<BR>\n" );
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document.write( "As a slight detour, let's find out how many students are in exactly one club only.<BR>\n" );
document.write( "artsOnly + academicsOnly + musicOnly = a + c + g = 6+2+3 = 11\r<BR>\n" );
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document.write( "There are 11 students who have signed up for exactly one club.\r<BR>\n" );
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document.write( "There are 30 students total, so that must mean 30-11 = 19 students are in at least two clubs.\r<BR>\n" );
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document.write( "The events \"in exactly one club\" and \"in at least two clubs\" are complementary events. One or the other must happen, but not both at the same time.\r<BR>\n" );
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document.write( "Final Answer: <font color=red>19 students</font> who belong to at least two clubs.<BR>\n" );
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