document.write( "Question 1201973: In survey of 900 workers 600 owned houses,500 owned cars and 345 owned boat,300 owned cars and houses,250 owned houses and boats and 270 owned cars and boats 200 owned the three.find how many workers don't owned any of items, find how many workers owned only two of the items \n" ); document.write( "

Algebra.Com's Answer #836962 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "In survey of 900 workers 600 owned houses, 500 owned cars and 345 owned $\"highlight%28cross%28boat%29%29\"$ boats,
\n" ); document.write( "300 owned cars and houses,250 owned houses and boats and 270 owned cars and $\"highlight%28cross%28boat%29%29\"$ boats.
\n" ); document.write( "200 owned the three.
\n" ); document.write( "(a) Find how many workers don't owned any of items.
\n" ); document.write( "(b) Find how many workers owned only two of the items.
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document.write( "(a)  You are given a universal set of 900 workers and three its basic subsets\r\n" );
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document.write( "         - H (owned houses) of 600 workers;\r\n" );
document.write( "         - C (owned cars)   of 500 workers;\r\n" );
document.write( "         - B (owned boats)  of 345 workers.\r\n" );
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document.write( "     You also are given the info about in-pair intersections of these three subsets\r\n" );
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document.write( "         - CH (owned cars   and houses)  of 300 workers;\r\n" );
document.write( "         - HB (owned houses and boats)   of 250 workers;\r\n" );
document.write( "         - CB (owned cars   and boats)   of 270 workers.\r\n" );
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document.write( "     You also are given the info about the triple intersections of these three subsets\r\n" );
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document.write( "         - CHB (owned all three items)  of 200 workers.\r\n" );
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document.write( "     Use the Inclusion-Exlusion principle.  The Inclusion-Exclusion formula is\r\n" );
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document.write( "         n(H U C U B) = n(H) + n(C) + n(B) - n(HC) - n(HB) - n(CB) + n (CHB) = \r\n" );
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document.write( "                      = 600 + 500 + 345 - 300 - 250 - 270 + 200 = 825.\r\n" );
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document.write( "     Thus 825 workers own at least one of three items.\r\n" );
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document.write( "     Hence, the rest  1000-825 = 175  workers don't have any of items.    ANSWER\r\n" );
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document.write( "(b)  To find the number of workers who own only two of the items, subtract triple intersection from \r\n" );
document.write( "     the corresponding in-pair intersection.\r\n" );
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document.write( "     You will get:\r\n" );
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document.write( "         n(CH_only) = n(CH) - n(CHB) = 300 - 200 = 100  (owned cars   and houses only);\r\n" );
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document.write( "         n(HB_only) = n(HB) - n(CHB) = 250 - 200 =  50  (owned houses and boats only);\r\n" );
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document.write( "         n(CB_only) = n(CB) - n(CHB) = 270 - 200 =  70  (owned cars   and boats only).\r\n" );
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document.write( "     These sets CH_only, HB_only and CB_only are disjoint (i.e. have empty intersection;  THEREFORE\r\n" );
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document.write( "     the number of workers owned only two of the items is the sum  100 + 50 + 70 = 220.    ANSWER\r\n" );
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\n" ); document.write( "\n" ); document.write( "To see many other similar (and different) solved problems on Inclusion-Exclusion principle, 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
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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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