document.write( "Question 1163106: From a group of 440 students who took the exam, 300 passed English, 280 passed Spanish, 220 passed Math, 180 passed English and Spanish, 160 Spanish and Mathematics, 150 English and Mathematics and 100 passed English, Spanish and Mathematics. How many altogether passed at least one of these three subjects? \n" ); document.write( "
Algebra.Com's Answer #787081 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "You are given the sets \r\n" );
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document.write( "    U - universal set of all 440 students\r\n" );
document.write( "    E - set of students passed English (300)\r\n" );
document.write( "    S - set of students passed Spanish (280)\r\n" );
document.write( "    M - set of students passed Math    (220)\r\n" );
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document.write( "the in-pair intersections\r\n" );
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document.write( "    ES - English and Spanish           (180)\r\n" );
document.write( "    SM - Spanish and Math              (160)\r\n" );
document.write( "    EM - English and Math              (150)\r\n" );
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document.write( "and the triple intersection\r\n" );
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document.write( "    ESM - English, Spanish and Math    (100)\r\n" );
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document.write( "For the union E U S U M, there is a REMARKABLE formula from elementary set theory \r\n" );
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document.write( "    n(E U S U M) = n(E) + n(S) + n(M) - N(ES) - n(SM) - n(EM) + n(ESM).    (1)\r\n" );
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document.write( "It says the number of elements in the union of three subsets is the alternate sum of the shown components.\r\n" );
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document.write( "By substituting the given values, you get the answer to the problem's question\r\n" );
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document.write( "    n(E U S U M) = 300 + 280 + 220 - 180 - 160 - 150 + 100 = 410.\r\n" );
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document.write( "ANSWER.  In all, 410 students passed at least one exam.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Memorize the formula and the method of the solution (!)\r
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\n" ); document.write( "\n" ); document.write( "The last step is to prove the formula (1).\r
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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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\n" ); document.write( "\n" ); document.write( "Thus the formula  (1)  is proved   //   and the solution is fully completed  (! !)\r
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\n" ); document.write( "\n" ); document.write( "To see many other similar solved problems, look into the lessons\r
\n" ); document.write( "\n" ); document.write( "    - Counting elements in sub-sets of a given finite set\r
\n" ); document.write( "\n" ); document.write( "    - Advanced problems on counting elements in sub-sets of a given finite set\r
\n" ); document.write( "\n" ); document.write( "    - Challenging problems on counting elements in subsets of a given finite set \r
\n" ); document.write( "\n" ); document.write( "    - Selected problems on counting elements in subsets of a given finite set \r
\n" ); document.write( "\n" ); document.write( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "Look also into the links\r
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\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/sets-and-operations/sets-and-operations.faq.question.1149313.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1132870.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1129554.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Probability-and-statistics/Probability-and-statistics.faq.question.1129554.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Rate-of-work-word-problems/Rate-of-work-word-problems.faq.question.1126097.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/word/evaluation/Evaluation_Word_Problems.faq.question.1126099.html\r
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\n" ); document.write( "\n" ); document.write( "to similar solved problems in the archive of this forum.\r
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\n" ); document.write( "\n" ); document.write( "To tutor @greenestamps:\r
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\n" ); document.write( "\n" ); document.write( "            Thanks for noticing my typos --- I just fixed them (!)\r
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