document.write( "Question 721538: Upon examining the contents of 38 backpacks, it was found that 23 contained a black pen, 27 contained a blue pen, and 21 contained a pencil, 15 contained both a black pen and a blue pen, 12 contained both a black pen and a pencil, 18 contained both a blue pen and a pencil, and 10 contained all three items. How many backpacks contained none of the three writing instruments? \n" ); document.write( "
Algebra.Com's Answer #442422 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "There are 3 overlapping sets A,B, and C inside the big rectangle,\r\n" );
document.write( "making 8 regions to consider.   \r\n" );
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document.write( "Circle A contains all the backpacks that had at least a black pen.\r\n" );
document.write( "Circle B contains all the backpacks that had at least a blue pen.\r\n" );
document.write( "Circle C contains all the backpacks that had at least a pencil.\r\n" );
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document.write( "We want to know how many are in\r\n" );
document.write( "region k, which is the region outside all three circles.\r\n" );
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document.write( "Let's reverse the order the clues\r\n" );
document.write( "are listed in:\r\n" );
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\n" ); document.write( "10 contained all three items.
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document.write( "So we put 10 in region f\r\n" );
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document.write( "

\n" ); document.write( "18 contained both a blue pen and a pencil,
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document.write( "We have accounted for 10 of these among the 18, so the remaining\r\n" );
document.write( "8 are in region i.  So we put 8 in i \r\n" );
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document.write( "

\n" ); document.write( "12 contained both a black pen and a pencil,
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document.write( "We have accounted for 10 of these among the 12, so the remaining\r\n" );
document.write( "2 are in region g.  So we put 2 in g\r\n" );
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document.write( "

\n" ); document.write( "15 contained both a black pen and a blue pen,
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document.write( "We have accounted for 10 of these among the 15, so the remaining\r\n" );
document.write( "5 are in region e.  So we put 5 in region e:\r\n" );
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\n" ); document.write( "21 contained a pencil,
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document.write( "We have filled in 3 of the regions of the bottom circle \r\n" );
document.write( "representing all the backpacks which had at least a pencil.\r\n" );
document.write( "2+10+8=20, so the remaining 1 of the 21 is in region j.\r\n" );
document.write( "So we put 1 in region j.\r\n" );
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document.write( "

\n" ); document.write( "27 contained a blue pen
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\n" ); document.write( "We have filled in 3 of the regions of the right circle
\n" ); document.write( "representing all the backpacks which had at least a blue pen.
\n" ); document.write( "5+10+8=23, so the remaining 4 of the 27 is in region h. So
\n" ); document.write( "we put 4 in region h.\r
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\n" ); document.write( "23 contained a black pen,
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document.write( "We have filled in 3 of the regions of the left circle \r\n" );
document.write( "representing all the backpacks which had at least a black pen.\r\n" );
document.write( "5+10+2=17, so the remaining 6 of the 23 is in region h.  So\r\n" );
document.write( "we put 6 in region d.\r\n" );
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document.write( "Now we come to the question:\r\n" );
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\n" ); document.write( "How many backpacks contained none of the three writing instruments?
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document.write( "Now we have the number in every region but k.  We are told\r\n" );
document.write( "there are 38 backpacks.  The other 7 regions contain\r\n" );
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document.write( "10+8+2+5+1+4+6 = 36\r\n" );
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document.write( "So the remining 2 are in the outside region k.  So we\r\n" );
document.write( "put 2 in region k:\r\n" );
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document.write( "Answer: 2\r\n" );
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document.write( "Edwin
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