document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1149313>Question 1149313</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a class of 36 students 25 study chemistry and 22 study mathematics and 25<BR>\n" );
document.write( "study physics ,17 study physics and mathematics,18 study physics and<BR>\n" );
document.write( "chemistry,15 study one of the three subject Find a)Those who study all three<BR>\n" );
document.write( "subject b) only mathematics and chemistry  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #770713 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52781)</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=770713\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "In a class of 36 students 25 study chemistry and 22 study mathematics and 25 study physics ,17 study physics and mathematics,<BR>\n" );
document.write( "18 study physics and chemistry,15 study one of the three subject <BR>\n" );
document.write( "Find a)Those who study all three subject b) only mathematics and chemistry <BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The solution by @greenestamps is correct;     &nbsp;&nbsp;&nbsp;&nbsp;(G)\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The solution by @math_helper is correct;      &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(MH)\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The solution by Edwin is &nbsp;NOT &nbsp;CORRECT.         &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(E)\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;My solution is &nbsp;BELOW.  &nbsp;It supports &nbsp;(G) &nbsp;and &nbsp;(MH) &nbsp;and disproves &nbsp;(E).\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;See also my comments after my solution.\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "We are given a universal set of 36 students (the entire class).\r\n" );
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document.write( "We are also given info about three its subsets : \r\n" );
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document.write( "    C (students enrolled to Chemistry)  of n(C) = 25 elements;  \r\n" );
document.write( "    M (Mathematics)                     of n(M) = 22 elements, and \r\n" );
document.write( "    P (Physics)                         of n(P) = 25 elements.\r\n" );
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document.write( "We are also given the following info about their in-pair intersections:\r\n" );
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document.write( "    PM (studying Physics and Math)      of n(PM) = 17 elements;\r\n" );
document.write( "    PC (studying Physics and Chemistry) of n(PC) = 18 elements;\r\n" );
document.write( "    MC (studying Math    and Chemistry) of n(MC) = unknown number of elements.\r\n" );
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document.write( "We also should find the number of elements in the the triple intersection\r\n" );
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document.write( "    PMC (studying Physics, Math and Chem) of n(PMC) (unknown number of elements).\r\n" );
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document.write( "We assume (from the <U>context</U>) that the union of subsets P, M and C  is the entire set of the 36 students in the class.\r\n" );
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document.write( "In this situation the following formula works\r\n" );
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document.write( "    n(P U M U C) = n(P) + n(M) + n(C) - n(P n M) - n(P n C) - n(M n C) + n(P n M n C).     (1)\r\n" );
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document.write( "Substitute the given data. You will get\r\n" );
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document.write( "    36 = 25 + 22 + 25 - 17 - 18 - n(MC) + n(PMC),   or\r\n" );
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document.write( "    n(MC) - n(PMC) = 25 + 22 + 25 - 17 - 18 - 36 = 1.\r\n" );
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document.write( "This difference, n(MC) - n(PMC), is exactly the number of those who study only Math and Chemistry,\r\n" );
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document.write( "so it is just answer to question b) of the problem.\r\n" );
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document.write( "Now we will work to answer question a) and to find the unknown n(PMC).\r\n" );
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document.write( "For it, I should create the other equation, based on the second given condition.\r\n" );
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document.write( "The number of those who study only Math      is  22-n(PM)-n(MC)+n(PMC) = 22-n(PM)-(n(MC)-n(PMC)) =\r\n" );
document.write( "                        replace here n(PM) by 17 and replace (n(MC)-n(PMC)) by 1, as we just found, to continue the line\r\n" );
document.write( "                                                                                                 = 22-17-1 = 4.\r\n" );
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document.write( "The number of those who study only Physics   is  25-n(PM)-n(PC)+n(PMC) = 25-17-18+n(PMC) = -10+n(PMC).\r\n" );
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document.write( "The number of those who study only Chemistry is  25-n(PC)-n(MC)+n(PMC) = 25-n(PC)-(nMC)-n(PMC) = \r\n" );
document.write( "                        replace here n(PC) by 18 and replace (n(MC)-n(PMC)) by 1, as we just found, to continue the line                   \r\n" );
document.write( "                                                                                               = 25-18-1 = 6.\r\n" );
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document.write( "Now the equation \"the sum of those who study only one of the three subjects\" is\r\n" );
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document.write( "    4 - 10+n(PMC) + 6 = 15,  which implies  n(PMC) = 15 - 4 + 10 - 6 = 15.\r\n" );
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document.write( "<U>ANSWER to (a)</U> : 15.    <U>ANSWER to (b)</U> : 1.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Solved.\r<BR>\n" );
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document.write( "The last step is to prove the formula (1).\r<BR>\n" );
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document.write( "    It is totally clear to you why I add the first three addends in the formula (1).\r\n" );
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document.write( "    But when I add them, I count twice every term in each in-pair intersection.\r\n" );
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document.write( "    Therefore, I subtract the numbers of terms in each in-pair intersection.\r\n" );
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document.write( "    Next, when I add three first addends, I count thrice each term in the triple intersection;\r\n" );
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document.write( "    and when I subtract in-pair intersections, I cancel these terms thrice.\r\n" );
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document.write( "    Therefore, I must add the number of terms in the triple intersection one more time to restore the balance.\r\n" );
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document.write( "Thus the formula &nbsp;(1) &nbsp;is proved &nbsp;&nbsp;// &nbsp;&nbsp;and the solution is completed &nbsp;(!&nbsp;!)\r<BR>\n" );
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document.write( "My comments.\r<BR>\n" );
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document.write( "    Probably, the formulation is not perfect; probably, it could be better.\r\n" );
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document.write( "    But as formulated, it does not leave any doubt/doubts\r\n" );
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document.write( "        - that the condition covers and involves all 36 students of the class, and\r\n" );
document.write( "        - what does it mean \"15 study one of the three subject\".\r\n" );
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document.write( "     It is a STANDARD WAY to formulate such circumstances in this sort of problems,\r\n" );
document.write( "     and I saw tens of them in this forum, formulated in this way.\r\n" );
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document.write( "     Regarding matrix solutions by Edwin and by @Math_helper, I do not think that it is the expected way for the school students\r\n" );
document.write( "     to get the solution.\r\n" );
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document.write( "     It is why I produced my own solution and placed it here.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Thank you for your attention (!)\r<BR>\n" );
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document.write( "===================\r<BR>\n" );
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document.write( "Dear reader, if you want to see other similar solved problems, to wider your horizon and to feel <BR>\n" );
document.write( "the solid ground under your legs, look into the lessons\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( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/word/misc/Challenging-problems-on-counting-elements-in-subsets-of-a-given-finite-set.lesson>Challenging problems on counting elements in subsets of a given finite set</A> \r<BR>\n" );
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document.write( "in this site.\r<BR>\n" );
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document.write( "Look also into the links\r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1132870.html\r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1129554.html\r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1129554.html\r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/Rate-of-work-word-problems/Rate-of-work-word-problems.faq.question.1126097.html\r<BR>\n" );
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document.write( "https://www.algebra.com/algebra/homework/word/evaluation/Evaluation_Word_Problems.faq.question.1126099.html\r<BR>\n" );
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document.write( "to similar solved problems in the archive of this forum.\r<BR>\n" );
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document.write( "When you learn from these sources, you will know the subject as well, as I know it . . . \r<BR>\n" );
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document.write( " </SPAN>\n" );
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