document.write( "Question 1190277: A survey of a group of 112 tourists was taken in St louis, the survey showed the following:
\n" ); document.write( "62 the Taurus plan to visit gateway Arch;
\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( "

Algebra.Com's Answer #821888 by ikleyn(52787)\"\" \"About 
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 \"highlight%28cross%28Taurus%29%29\" tourists plan to visit gateway Arch;
\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 \"n%28M%5Bonly%5D%29\",  which is \r\n" );
document.write( "\r\n" );
document.write( "    \"n%28M%5Bonly%5D%29\" = n(M) - ( n(AM) + n(MZ) - n(AMZ) ) = 44 - (20 + 19 - 8) = 13.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "ANSWER.  13 tourists plan to visit the museum only.\r\n" );
document.write( "
\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( "
\n" );