document.write( "Question 1095656: Dear Tutor,
\n" ); document.write( "Below I have tried to solve the following problem, but with no avail. I would greatly appreciate your help, please: \r
\n" ); document.write( "\n" ); document.write( "When a group of colleagues discussed where their annual retreat should take​ place they found that of all the​ colleagues: 10 would not go to a​ park 17 would not go to a​ beach 12 would not go to a​ cottage 3 would go to neither a park nor a​ beach 11 would go to neither a beach nor a​ cottage 2 would go to neither a park nor a​ cottage 1 would not go to a park or a beach or a​ cottage and 2 were willing to go to all three places. What is the total number of colleagues in the​ group?\r
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\n" ); document.write( "Jen B \n" ); document.write( "
Algebra.Com's Answer #710167 by ikleyn(52786)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Let  A  be the subset of those 10 who would not go to a​ park;\r\n" );
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document.write( "let  B  be the subset of those 17 who would not go to a​ beach;  and\r\n" );
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document.write( "let  C  be the subset of those 12 who would not go to a​ cottage.\r\n" );
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document.write( "Then we are given that \r\n" );
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document.write( "    - the intersection AB of the subsets A and B  consists of 3 persons (=\"neither a park nor a​ beach\");\r\n" );
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document.write( "    - the intersection BC of the subsets B and C  consists of 11 persons (=\"neither a beach nor a​ cottage\");\r\n" );
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document.write( "    - the intersection AC of the subsets A and C  consists of 2 persons (=\"neither a park nor a​ cottage\").\r\n" );
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document.write( "We are also given that the intersection ABC of the subsets A, B and C consists of 1 person (\"would not go to a park or a beach or a​ cottage\"),   and\r\n" );
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document.write( "the supplement of the UNION of the sets A, B and C to the entire group consists of 2 persons (\"willing to go to all three places\").\r\n" );
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document.write( "Now, there is a FUNDAMENTAL and ELEMENTARY formula in the theory of finite sets saying that\r\n" );
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document.write( "    n(A U B U C) = nA + nB + nC - nAB - nBC - nAC + nABC.\r\n" );
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document.write( "Here the small letter n before the set/the subset name means \"the number of elements in the subset\", \r\n" );
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document.write( "    or, using the high level terminology, \"the cardinality\" of the finite subset.\r\n" );
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document.write( "I will not distract your attention now for proving this formula.\r\n" );
document.write( "Instead, I will show you how to solve the problem in two lines, using this formula.\r\n" );
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document.write( "    Line 1:  n(A U B U C) = 10 + 17 + 12 - 3 - 11 - 2 + 1 = 24 persons in the UNION  (A U B U C),   and\r\n" );
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document.write( "    Line 2:  The entire set = (A U B U C) + 2 persons   has  24 + 2 = 26 persons in total.\r\n" );
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document.write( "Answer.  The entire group consists of 26 persons.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Instead of proving the formula  (*),  I'll direct you to my lessons in this site \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
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\n" ); document.write( "\n" ); document.write( "You will find there all the arguments needed for the proof  (and, actually, the proof itself)  in the entertainment form.\r
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\n" ); document.write( "\n" ); document.write( "            Read it and have fun !\r
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