document.write( "Question 1158213: Among 18 students in a room, 7 study mathematics, 10 study science, and 10 study computer
\n" ); document.write( "programming. Also, 3 study mathematics and science, 4 study mathematics and computer
\n" ); document.write( "programming, and 5 study science and computer programming. We know that 1 student studies all three
\n" ); document.write( "subjects. How many of these students study none of the three subjects?
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #781150 by ikleyn(52781)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "Among 18 students in a room, 7 study mathematics, 10 study science, and 10 study computer
\n" ); document.write( "programming. Also, 3 study mathematics and science, 4 study mathematics and computer
\n" ); document.write( "programming, and 5 study science and computer programming. We know that 1 student studies all three
\n" ); document.write( "subjects. How many of these students study none of the three subjects?
\n" ); document.write( "~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "Let M be the set of those who study (at least) Math     ( n(M) = 7 )\r\n" );
document.write( "\r\n" );
document.write( "Let S be the set of those who study (at least) Science  ( n(S) = 10 )\r\n" );
document.write( "\r\n" );
document.write( "Let C be the set of those who study (at least) Computer ( n(C) = 10 )\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Let MS, MC and SC be the corresponding intersetion sets (those who study at least two subjects)\r\n" );
document.write( "\r\n" );
document.write( "Let MSC be the intersection set (M & S & C).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "There is a REMARCABLE formula in elementary set theory\r\n" );
document.write( "\r\n" );
document.write( "    n(M U S U C) = n(M) + n(S) + n(C) - n(MS) - n(MC) - n(SC) + n(MSC).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It allows calculating the number of elements in the UNION of separate sub-sets.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In our case, this formula gives the value\r\n" );
document.write( "\r\n" );
document.write( "    n(M U S U C) = 7 + 10 + 10 - 3 - 4 - 5 + 1 = 16.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Thus 16 students study at least one of listed subjects.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Hence, the rest 18-16 = 2 in the room study none of the three subjects.    ANSWER\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "--------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To learn more about this formula, see the lessons\r
\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( "in this site.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "You will find there a variety of other similar solved problems.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );