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\n" ); document.write( "Resorts. In a survey of 65 resorts, it was found that
\n" ); document.write( "34 provided refrigerators in the guest rooms
\n" ); document.write( "30 provided laundry services
\n" ); document.write( "37 provided child care services
\n" ); document.write( "15 provided refrigerators in the guest rooms and laundry services
\n" ); document.write( "17 provided refrigerators in the guest rooms and child care services
\n" ); document.write( "19 provided laundry services and child care services
\n" ); document.write( " 7 provided all three features
\n" ); document.write( "How many of the resorts provided
\n" ); document.write( "a. only refrigerators in the guest rooms?
\n" ); document.write( "b, exactly one of the features?
\n" ); document.write( "c. at least one of the features?
\n" ); document.write( "d. exactly two of the features?
\n" ); document.write( "e. none of the features?
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document.write( "In this problem, you are given a universal set of 65 resorts,\r\n" );
document.write( "and 3 (three) its subsets\r\n" );
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document.write( " R = {provided refrigerators in the guest rooms}; n(R) = 34;\r\n" );
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document.write( " L = {provided laundry services}; n(L) = 30;\r\n" );
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document.write( " C = {provided child care services}; n(C) = 37.\r\n" );
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document.write( "You are also given the number of elements in each in-pair intersetion\r\n" );
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document.write( " n(RL) = 15; n(RC) = 17; n(LC) = 19,\r\n" );
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document.write( "and in the triple intersection n(RLC) = 7.\r\n" );
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document.write( " +-----------------------------------------------+\r\n" );
document.write( " | Now I answer questions, one after other. |\r\n" );
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document.write( "(a) n(R_only) = n(R) - n(RL) - n(RC) + n(RLC) = 34 - 15 - 17 + 7 = 9.\r\n" );
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document.write( "(b) To answer (b), calculate n(L_only) and n(C_only similarly.\r\n" );
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document.write( " Then compute n(R_only) + n(L_only) + n(C_only).\r\n" );
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document.write( " I leave it to you, since after my instructions, it is just simple arithmetic.\r\n" );
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document.write( "(c) Use the inclusion-exlusion principle formula for the union (R U L U C)\r\n" );
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document.write( " n(R U L U C) = n(R) + n(L) + n(C) - n(RL) - n(RC) - n(LC) + n(RLC) = \r\n" );
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document.write( " = 34 + 30 + 37 - 15 - 17 - 19 + 7 = 57.\r\n" );
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document.write( "(e) is the complement of n(R U L U C) to 65, so the answer to (e) is 65 - 57 = 8.\r\n" );
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document.write( "(d) To answer (d), you should calculate n(RL_only), n(RC_only) and n(LC_only).\r\n" );
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document.write( " Each of these three numbers is calculated following the same scheme. \r\n" );
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document.write( " For example, n(RL_only) = n(RL) - n(RLC) = 15-7 = 8.\r\n" );
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document.write( " So, calculate n(RC_only) and n(LC_only) similarly;\r\n" );
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document.write( " then compute the sum n(RL_only) + n(RC_only) + n(LC_only).\r\n" );
document.write( "\r\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "To see many other similar (and different) solved problems, see the lessons\r
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\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( "in this site.\r
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\n" ); document.write( "\n" ); document.write( "All these problems relate to inclusion-exclusion principle.\r
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\n" ); document.write( "\n" ); document.write( "On inclusion-exclusion principle, see this Wikipedia article\r
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\n" ); document.write( "\n" ); document.write( "https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle\r
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