document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1085902>Question 1085902</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Dear Tutor,<BR>\n" );
document.write( "please help me.\r<BR>\n" );
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document.write( "20 students take a test with three questions A,B and C. 16 students answer at least one question correctly. 10 answer question A correctly, 8 answer question B correctly, 6 answer question C correctly. 3 students answer both A and B correctly, 4 answer A and C correctly and 1 student answers all three questions correctly. How many students answer questions B and C correctly?\r<BR>\n" );
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document.write( "Could you please explain steps taken while solving this problem?\r<BR>\n" );
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document.write( "Thank you. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #700052 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52803)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><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=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=700052\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
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document.write( "Let \"a\" be the set of students that answered the question A correctly.\r\n" );
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document.write( "Let \"b\" be the set of students that answered the question B correctly.\r\n" );
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document.write( "Let \"c\" be the set of students that answered the question C correctly.\r\n" );
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document.write( "Let ab be the set of students that answered both question A and B correctly.\r\n" );
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document.write( "Let bc be the set of students that answered both question B and C correctly.\r\n" );
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document.write( "Let ac be the set of students that answered both question A and C correctly.\r\n" );
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document.write( "Let abc be the set of students that answered all three question A, B and C correctly.\r\n" );
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document.write( "Next, for any finite set \"S\" in this problem i will denote by |S| the number of its elements.\r\n" );
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document.write( "From the elementary set theory, this formula is well known\r\n" );
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document.write( "|a U b U c| = |a| + |b| + |c| - |ab| - |ac| - |ab| + |abc|    (1)    (see the reference at the end of my post)\r\n" );
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document.write( "In our case  |a U b U c| = 16: it is the set of students who correctly answered at least one question.\r\n" );
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document.write( "Further, from the condition\r\n" );
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document.write( "|a| = 10,  |b| = 8,  |c| = 6,  |ab| = 3,  |ac| = 4,  |abc| = 1.\r\n" );
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document.write( "The value |bc| is unknown, and we EASILY will find it from the equation (1) after substituting all other known values to the equation:\r\n" );
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document.write( "16 = 10 + 8 + 6 - 3 - 4 - |bc| + 1.\r\n" );
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document.write( "It gives |bc| = 10 + 8 + 6 - 3 - 4 + 1 - 16 = 2.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "<U>Answer</U>.  2 students answered correctly questions B and C.\r<BR>\n" );
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document.write( "Regarding the formula (1), read these two lessons in this site\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/word/misc/Counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Counting elements in sub-sets of a given finite set</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/word/misc/Advanced-probs-counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Advanced problems on counting elements in sub-sets of a given finite set</A>\r<BR>\n" );
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document.write( "Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.\r<BR>\n" );
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document.write( "The referred lessons are the part of this online textbook under the topic <BR>\n" );
document.write( "\"<U>Miscellaneous word problems</U>\".\r<BR>\n" );
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document.write( "Ignore everything the tutor \"Theo\" wrote in his post: it is not relevant . . . Unfortunately.\r<BR>\n" );
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document.write( "-------------<BR>\n" );
document.write( "Usually, I read attentively the students' posts, as well as the tutors' posts.\r<BR>\n" );
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document.write( "(Except statistic problems, where I am not an expert).\r<BR>\n" );
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document.write( "But I never read the tutor's Theo posts: they always are so long that are simply unreadable.\r<BR>\n" );
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document.write( "I do not believe that the right Math problem may have long formulation. \r<BR>\n" );
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document.write( "I also do not believe that the right solution to elementary Math problem should be long.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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