document.write( "Question 1190277: A survey of a group of 112 tourists was taken in St louis, the survey showed the following:
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document.write( "62 the Taurus plan to visit gateway Arch;
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document.write( "50 plan to visit the zoo;
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document.write( "11 plan to visit the art museum and the zoo, but not the gateway Arch;
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document.write( "12 plan to visit the art museum and the gateway arch, but not the zoo;
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document.write( "19 plan to visit the gateway Arch and the zoo, but not the art museum;
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document.write( "8 plan to visit the art museum, the zoo, and the gateway Arch;
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document.write( "14 plan to visit none of the three places.
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document.write( "How many plan to visit the art museum only? \n" );
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Algebra.Com's Answer #821888 by ikleyn(52787)![]() ![]() You can put this solution on YOUR website! . \n" ); document.write( "A survey of a group of 112 tourists was taken in St Louis, the survey showed the following: \n" ); document.write( "62 the \n" ); document.write( "50 plan to visit the zoo; \n" ); document.write( "11 plan to visit the art museum and the zoo, but not the gateway Arch; \n" ); document.write( "12 plan to visit the art museum and the gateway arch, but not the zoo; \n" ); document.write( "19 plan to visit the gateway Arch and the zoo, but not the art museum; \n" ); document.write( "8 plan to visit the art museum, the zoo, and the gateway Arch; \n" ); document.write( "14 plan to visit none of the three places. \n" ); document.write( "How many plan to visit the art museum only? \n" ); document.write( "~~~~~~~~~~~~~~\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \r\n" ); document.write( "We have a universal set U of 112 tourists (total) and 3 its subsets\r\n" ); document.write( "\r\n" ); document.write( " A wishing to visit Arch ( n(A) = 62 )\r\n" ); document.write( "\r\n" ); document.write( " M wishing to visit Museum ( n(M) = ? )\r\n" ); document.write( "\r\n" ); document.write( " Z wishing to visit Zoo. ( n(Z) = 50 )\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "We know n(A) = 62,\r\n" ); document.write( "\r\n" ); document.write( " n(Z) = 50.\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "We have some info about in-pair intersections of these subsets AM, AZ and MZ, as well as about \r\n" ); document.write( "their triple intersection AMZ. This info is\r\n" ); document.write( "\r\n" ); document.write( " n(MZ \ AMZ) = 11,\r\n" ); document.write( "\r\n" ); document.write( " n(AM \ AMZ) = 12,\r\n" ); document.write( "\r\n" ); document.write( " n(AZ \ AMZ) = 19,\r\n" ); document.write( "\r\n" ); document.write( " n(AMZ) = 8.\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "From this info we have n(MZ) = 11+8 = 19; n(AM) = 12+8 = 20; n(AZ) = 19+8 = 27.\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Having it, we can apply the inclusive-exclusive formula (principle), which says\r\n" ); document.write( "\r\n" ); document.write( " n(A U M U Z) = n(A) + n(M) + n(Z) - n(AM) - n(AZ) - n(MZ) + n(AMZ).\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "Next, substitute what you just know into the formula. Notice that n(A U M U Z) = 112-14 = 98.\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "At this moment, we know all the terms in this formula, except of n(M)\r\n" ); document.write( "\r\n" ); document.write( " 98 = 62 + n(M) + 50 - 20 - 19 - 27 + 8, or 98 = 54 + n(M).\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "From the last equation, we get n(M) = 98 - 54 = 44.\r\n" ); document.write( "\r\n" ); document.write( "\r\n" ); document.write( "The problem asks us about\r \n" ); document.write( "\n" ); document.write( "Solved.\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "--------------------\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "On inclusion-exclusion principle, see this Wikipedia article\r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle\r \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "To see many other similar (and different) solved problems, see the lessons\r \n" ); document.write( " \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( " - Inclusion-Exclusion principle problems \r \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "in this site.\r \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( "Happy learning (!)\r \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "\n" ); document.write( " \n" ); document.write( " |