document.write( "Question 736918: a survey of 65 people was conducted during a seafood night in a local community.25 people like lobster,30 like fish and 19 like whelks.4 liked lobster and whelks only,2 liked whelks and fish only and 3 liked fish and lobster only.6 people did not like any of the 3 choices.\r
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Algebra.Com's Answer #852697 by ikleyn(53427) $\"\"$ $\"About$

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\n" ); document.write( "a survey of 65 people was conducted during a seafood night in a local community.
\n" ); document.write( "25 people like lobster, 30 like fish and 19 like whelks.
\n" ); document.write( "4 liked lobster and whelks only, 2 liked whelks and fish only and 3 liked fish and lobster only.
\n" ); document.write( "6 people did not like any of the 3 choices.
\n" ); document.write( "Determine the number of people who liked all three choices.
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document.write( "In this problem, we have an \"universal\" set of 65 people and several its subsets.\r\n" );
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document.write( "Subset L of 25 people who like lobster;\r\n" );
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document.write( "Subset F of 30 people who like fish;\r\n" );
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document.write( "Subset W of 19 people who like whelks;\r\n" );
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document.write( "Subset LWo of 4 persons who like lobster and whelks only;\r\n" );
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document.write( "Subset WFo of 2 persons who like whelks and fish only;\r\n" );
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document.write( "Subset FLo of 3 persons who like fish and lobster and only;\r\n" );
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document.write( "Subset N of 6 persons who did not like any of the 3 choices.\r\n" );
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document.write( "Let LFW be the subset of people who like all three choices, and let x be the number of people in this subset.\r\n" );
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document.write( "Then (LWo U LWF) is the disjoint union, equal to the intersection LW with (4+x) members;\r\n" );
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document.write( "     (WFo U LWF) is the disjoint union, equal to the intersection WF with (2+x) members;\r\n" );
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document.write( "     (FLo U LWF) is the disjoint union, equal to the intersection FL with (3+x) members.\r\n" );
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document.write( "Now apply the Inclusive-Exclusive principle, which says\r\n" );
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document.write( "    |(L U F U W)| = |L| + |F| + |W| - |LW| - |WF| - |FL| + |LFW|\r\n" );
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document.write( "Substitute here\r\n" );
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document.write( "    |(L U F U W)| = 65 - 6 = 59;\r\n" );
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document.write( "    |L| = 25,  |F| = 30,  |W| = 19,  |LW| = 4+x,  |WF| = 2+x,  |FL| = 3+x,  |LFW| = x.\r\n" );
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document.write( "Equation (1) becomes\r\n" );
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document.write( "    59 = 25 + 30 + 19 - (4+x) - (2+x) - (3+x) + x,\r\n" );
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document.write( "    59 = 65 - 2x,\r\n" );
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document.write( "    2x = 65 - 59 = 6,\r\n" );
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document.write( "     x           = 6/2 = 3.\r\n" );
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document.write( "ANSWER.  The number of those who like all 3 choices is 3.\r\n" );
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