document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1118582>Question 1118582</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The Department of Foreign Languages of a liberal arts college conducted a survey of its recent graduates to determine the foreign language courses they had taken while undergraduates at the college. Of the 530 graduates\r<BR>\n" );
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document.write( "208	had at least one year of Spanish.<BR>\n" );
document.write( "174	had at least one year of French.<BR>\n" );
document.write( "144	had at least one year of German.<BR>\n" );
document.write( "40	had at least one year of Spanish and French.<BR>\n" );
document.write( "29	had at least one year of Spanish and German.<BR>\n" );
document.write( "23	had at least one year of French and German.<BR>\n" );
document.write( "5	had at least one year of all three languages.\r<BR>\n" );
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document.write( "(a) How many of the graduates had at least 1 yr of at least one of the three languages?<BR>\n" );
document.write( "(b) How many of the graduates had at least 1 yr of exactly one of the three languages?<BR>\n" );
document.write( "c) How many of the graduates had less than 1 yr of any of the three languages? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #733977 by </B><A HREF=/tutors/aboutme.mpl?userid=ikleyn><B>ikleyn(52814)</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=733977\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> .<BR>\n" );
document.write( "The Department of Foreign Languages of a liberal arts college conducted a survey of its recent graduates to determine <BR>\n" );
document.write( "the foreign language courses they had taken while undergraduates at the college. Of the 530 graduates\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "208	had at least one year of Spanish.\r\n" );
document.write( "174	had at least one year of French.\r\n" );
document.write( "144	had at least one year of German.\r\n" );
document.write( "40	had at least one year of Spanish and French.\r\n" );
document.write( "29	had at least one year of Spanish and German.\r\n" );
document.write( "23	had at least one year of French and German.\r\n" );
document.write( "5	had at least one year of all three languages.\r\n" );
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document.write( "(a) How many of the graduates had at least 1 yr of at least one of the three languages?\r\n" );
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document.write( "(b) How many of the graduates had at least 1 yr of exactly one of the three languages?\r\n" );
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document.write( "c) How many of the graduates had less than 1 yr of any of the three languages?\r\n" );
document.write( "</PRE>~~~~~~~~~~~~~~~~~~~~~~~\r<BR>\n" );
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document.write( "You are given\r\n" );
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document.write( "S = 208\r\n" );
document.write( "F = 174\r\n" );
document.write( "G = 144\r\n" );
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document.write( "SF = 40\r\n" );
document.write( "SG = 29\r\n" );
document.write( "FG = 23\r\n" );
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document.write( "SFG = 5\r\n" );
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document.write( "(a)  How many of the graduates had at least 1 yr of at least one of the three languages?\r\n" );
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document.write( "     There is a special formula to calculate n(S U F U G), when you know the numbers of elements in the subsets S, F, G \r\n" );
document.write( "     and in each of intersections SF, SG, FG, SFG:\r\n" );
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document.write( "     n(S U F U G) = S   + F   + G   - SF - SG - FG + SFG = \r\n" );
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document.write( "                  = 208 + 174 + 144 - 40 - 29 - 23 + 5 = 439.\r\n" );
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document.write( "     <U>Answer</U> to question (a) is 439.\r\n" );
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document.write( "(b)  How many of the graduates had at least 1 yr of exactly one of the three languages?\r\n" );
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document.write( "     (S - SF - SG + SFG) + (F - SF - FG + SFG) + (G - SG - FG + SFG).\r\n" );
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document.write( "      Substitute the given data and calculate.\r\n" );
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document.write( "c)  How many of the graduates had less than 1 yr of any of the three languages?\r\n" );
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document.write( "    Answer to this question is the difference  530 - 439,\r\n" );
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document.write( "    where 530 came from the condition and 439 is the answer to question (a).\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "--------------------<BR>\n" );
document.write( "If you need more explanations, &nbsp;look into 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( " </SPAN>\n" );
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