document.write( "Question 1206820: A school sports team contains 68 students. 33 do field events, 40 do track events, 23 do swimming, 14 do both field and track events, 8 do both swimming and field events. If 15 students do field events only and 10 do both swimming and track events, how many students do
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Algebra.Com's Answer #844483 by ikleyn(52777) $\"\"$ $\"About$

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\n" ); document.write( "A school sports team contains 68 students.
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\n" ); document.write( "If 15 students do field events only and 10 do both swimming and track events, how many students do
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document.write( "We are given a universal set of 68 students and 3 its basic subsets \r\n" );
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document.write( "    F  of 33 students\r\n" );
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document.write( "    T  of 40 students\r\n" );
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document.write( "    S  of 23 students.\r\n" );
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document.write( "We also are given three in-pair intersections\r\n" );
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document.write( "    FT of 14 students\r\n" );
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document.write( "    FS of  8 students\r\n" );
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document.write( "    TS of 10 students.\r\n" );
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document.write( "We also are given that the set\r\n" );
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document.write( "    F_only contains 15 students.\r\n" );
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document.write( "Let x be the number of students that do all 3 activities FTS.\r\n" );
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document.write( "Then, due to the Inclusion-Exclusion principle, we have this equation\r\n" );
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document.write( "    68 - n(F) + n(T) + n(S) - n(FT) - n(FS) - n(TS) + x,\r\n" );
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document.write( "or\r\n" );
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document.write( "    68 = 33 + 40 + 23 - 14 - 8 - 10 + x.\r\n" );
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document.write( "From it, we finf\r\n" );
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document.write( "    x = 68 - 33 - 40 - 23 + 14 + 8 + 10 = 4.\r\n" );
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document.write( "So, the ANSWER to (c) is 4 students do all 3 activities.\r\n" );
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document.write( "Now for (a), \"swimming only\" has  S_only = S - SF - TS + FTS = 23 -  8 - 10 + 4 =  9  students.   ANSWER\r\n" );
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document.write( "Now for (b), \"track only\"    has  T_only = T - FT - TS + FTS = 40 - 14 - 10 + 4 = 20 students.    ANSWER\r\n" );
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document.write( "At this point, the problem is solved in full. \r\n" );
document.write( "All questions are answered.\r\n" );
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