document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1078665>Question 1078665</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A survey of 100 students at New England College showed the following:<BR>\n" );
document.write( "45		take English.<BR>\n" );
document.write( "47		take history.<BR>\n" );
document.write( "37		take language.<BR>\n" );
document.write( "13		take English and history.<BR>\n" );
document.write( "17		take English and language.<BR>\n" );
document.write( "19		take history and language.<BR>\n" );
document.write( "7		take all three.<BR>\n" );
document.write( "How many students do the following? (See Example 6.)<BR>\n" );
document.write( "(a) Take history but neither of the other two?<BR>\n" );
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document.write( "(b) Take English and history but not language?<BR>\n" );
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document.write( "(c) Take none of the three?<BR>\n" );
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document.write( "(d) Take just one of the three?<BR>\n" );
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document.write( "(e) Take exactly two of the three?<BR>\n" );
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document.write( "(f) Do not take language?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #693236 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52878)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ikleyn><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=ikleyn><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=693236\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A survey of 100 students at New England College showed the following:<BR>\n" );
document.write( "45		take English.<BR>\n" );
document.write( "47		take history.<BR>\n" );
document.write( "37		take language.<BR>\n" );
document.write( "13		take English and history.<BR>\n" );
document.write( "17		take English and language.<BR>\n" );
document.write( "19		take history and language.<BR>\n" );
document.write( "7		take all three.<BR>\n" );
document.write( "How many students do the following? (See Example 6.)<BR>\n" );
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document.write( "Let E = the set of students learning English, and let n(E)= 45 be the number of elements in this set.\r\n" );
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document.write( "Let H = the set of students learning History, and let n(H)= 47 be the number of elements in this set.\r\n" );
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document.write( "Let L = the set of students learning Language, and let n(L)= 37 be the number of elements in this set.\r\n" );
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document.write( "Let EH = the intersection of E and H, and let n(EH)= 13 be the number  of elements in this set.\r\n" );
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document.write( "Let EL = the intersection of E and L, and let n(EL)= 17 be the number  of elements in this set.\r\n" );
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document.write( "Let HL = the intersection of H and L, and let n(EL)= 19 be the number  of elements in this set.\r\n" );
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document.write( "Let EHL = the intersection of E, H and L, and let n(EHL)= 7 be the number  of elements in this set.\r\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;After introducing these notations, &nbsp;<U>we are ready</U> &nbsp;to &nbsp;<U>solve the problem</U> &nbsp;<U>EASILY</U>.\r<BR>\n" );
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document.write( "(a) Take history but neither of the other two?\r\n" );
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document.write( "        = n(H) - n(EH) - n(HL) + n(EHL) = 47 - 13 - 19 + 7 = 22.\r\n" );
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document.write( "(b) Take English and history but not language?\r\n" );
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document.write( "        = n(EH) - n(EHL) = 13 - 7 = 6.\r\n" );
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document.write( "(c) Take none of the three?\r\n" );
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document.write( "        = 100 - [n(E) + n(H) + n(L) - N(EH) - n(HL) - n(EL) + n(EHL)] = calculate it on your own: simply substitute data\r\n" );
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document.write( "                (What you see in BRACKETS is the number of those who takes at least one subject)\r\n" );
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document.write( "(d) Take just one of the three?\r\n" );
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document.write( "        = [n(E)-n(EH)-n(EL)+n(EHL)] + [n(L)-n(EL)-n(HL)+n(EHL)] + [n(H)-n(EH)-n(HL)+n(EHL)]] = calculate it on your own: simply substitute data\r\n" );
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document.write( "(e) Take exactly two of the three?\r\n" );
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document.write( "        = [n(EH) - n(EHL)] + [n(HL) - n(EHL)] + [n(EL) - n(EHL)] = calculate it on your own: simply substitute data\r\n" );
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document.write( "(f) Do not take language?\r\n" );
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document.write( "        = 100 - n(L) = 100 - 37 = 63.\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "See the lessons\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=http://www.algebra.com/algebra/homework/word/misc/Counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Counting elements in sub-sets of a given finite set</A>\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/word/misc/Advanced-probs-counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Advanced problems on counting elements in sub-sets of a given finite set</A>\r<BR>\n" );
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document.write( "in this site.\r<BR>\n" );
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document.write( "Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site\r<BR>\n" );
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document.write( "&nbsp;&nbsp;&nbsp;&nbsp;- <A  rel=nofollow HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.\r<BR>\n" );
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document.write( "The referred lessons are the part of this online textbook under the topic <BR>\n" );
document.write( "\"<U>Miscellaneous word problems</U>\".\r<BR>\n" );
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