document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191784>Question 1191784</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a group of 200 students, each student studies at least one of the three science subjects:<BR>\n" );
document.write( "Biology, Chemistry and Physics. 130 study Biology, 135 study Chemistry, 115 study Physics, 86<BR>\n" );
document.write( "study Biology and Chemistry, 70 study Chemistry and Physics, and 64 study Physics and Biology.<BR>\n" );
document.write( "Illustrate this information on a clearly labelled Venn diagram, showing the number of elements<BR>\n" );
document.write( "in each separate region. Hence find the number of students who study<BR>\n" );
document.write( "a.All 3 subjects<BR>\n" );
document.write( "b. Exactly 2 subjects<BR>\n" );
document.write( "c.only biology  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849107 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849107\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem using a Venn diagram:\r<BR>\n" );
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document.write( "**1. Set up the Venn Diagram:**\r<BR>\n" );
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document.write( "Draw three overlapping circles representing Biology (B), Chemistry (C), and Physics (P).\r<BR>\n" );
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document.write( "**2. Use the Principle of Inclusion-Exclusion:**\r<BR>\n" );
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document.write( "Let:<BR>\n" );
document.write( "* |B| = Number of students studying Biology = 130<BR>\n" );
document.write( "* |C| = Number of students studying Chemistry = 135<BR>\n" );
document.write( "* |P| = Number of students studying Physics = 115<BR>\n" );
document.write( "* |B ∩ C| = Number of students studying Biology and Chemistry = 86<BR>\n" );
document.write( "* |C ∩ P| = Number of students studying Chemistry and Physics = 70<BR>\n" );
document.write( "* |P ∩ B| = Number of students studying Physics and Biology = 64<BR>\n" );
document.write( "* |B ∩ C ∩ P| = Number of students studying all three subjects (what we need to find)\r<BR>\n" );
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document.write( "The Principle of Inclusion-Exclusion for three sets is:\r<BR>\n" );
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document.write( "|B ∪ C ∪ P| = |B| + |C| + |P| - |B ∩ C| - |C ∩ P| - |P ∩ B| + |B ∩ C ∩ P|\r<BR>\n" );
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document.write( "We know that all 200 students study at least one subject, so |B ∪ C ∪ P| = 200. Plugging in the values:\r<BR>\n" );
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document.write( "200 = 130 + 135 + 115 - 86 - 70 - 64 + |B ∩ C ∩ P|<BR>\n" );
document.write( "200 = 160 + |B ∩ C ∩ P|<BR>\n" );
document.write( "|B ∩ C ∩ P| = 40\r<BR>\n" );
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document.write( "**3. Fill in the Venn Diagram:**\r<BR>\n" );
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document.write( "* **B ∩ C ∩ P:** 40 (all three subjects)<BR>\n" );
document.write( "* **B ∩ C only:** 86 - 40 = 46<BR>\n" );
document.write( "* **C ∩ P only:** 70 - 40 = 30<BR>\n" );
document.write( "* **P ∩ B only:** 64 - 40 = 24<BR>\n" );
document.write( "* **B only:** 130 - 46 - 40 - 24 = 20<BR>\n" );
document.write( "* **C only:** 135 - 46 - 40 - 30 = 19<BR>\n" );
document.write( "* **P only:** 115 - 24 - 40 - 30 = 21\r<BR>\n" );
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document.write( "**4. Answers:**\r<BR>\n" );
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document.write( "* **a. All 3 subjects:** 40 students<BR>\n" );
document.write( "* **b. Exactly 2 subjects:** 46 + 30 + 24 = 100 students<BR>\n" );
document.write( "* **c. Only Biology:** 20 students<BR>\n" );
document.write( " </SPAN>\n" );
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