document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=949746>Question 949746</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 7 freshmen, 6 sophomores, 7 juniors, and 4 seniors are eligible to be on a committee.\r<BR>\n" );
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document.write( "In how many ways can a dance committee of 18 students be chosen if it is to consist of at least 1 freshmen, 1 sophomore, 2 juniors, 2 seniors, and the rest can be of any grade?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #579936 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=579936\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is just my interpretation. Judge for yourself, because I may be overlooking some mistake.<BR>\n" );
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document.write( "There are lots of ways to make a committee of 18 students,<BR>\n" );
document.write( "if you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B6%2B7%2B4=24 ALIGN=MIDDLE  ALT=\"7%2B6%2B7%2B4=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > eligible candidates.<BR>\n" );
document.write( "In fact there are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C24%2C18%29%29=%28matrix%282%2C1%2C24%2C6%29%29=24%2A23%2A22%2A21%2A20%2A19%2F%281%2A2%2A3%2A4%2A5%2A6%29=23%2A22%2A19%2A7%2A2=134596 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "possible groups of 18 people that can be selected from a pool 24 people,<BR>\n" );
document.write( "or from a different point of view,<BR>\n" );
document.write( "there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=134596 ALIGN=MIDDLE  ALT=\"134596\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible groups of 6 rejected candidates that you can send home.<BR>\n" );
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document.write( "However, some of those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=134596 ALIGN=MIDDLE  ALT=\"134596\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible committees will fail the requirement to consist of at least 1 freshmen, 1 sophomore, 2 juniors, and 2 seniors.<BR>\n" );
document.write( "How many of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=134596 ALIGN=MIDDLE  ALT=\"134596\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible committees fail the requirement, and how/why do they fail?<BR>\n" );
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document.write( "No committee can fail because of not including 1 freshman,<BR>\n" );
document.write( "because without the 7 freshmen, we have only<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24-7=17 ALIGN=MIDDLE  ALT=\"24-7=17\"   DISABLED_style= \"max-width:90%; height:auto;\" > eligible candidates, and that is not enough for an 18-member committee.<BR>\n" );
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document.write( "Can a committee of 18 fail to include 1 sophomore?<BR>\n" );
document.write( "Yes, because we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24-6=18+non-sophomores%2C%0D%0Aand+they+would+form+%7B%7B%7B1 ALIGN=MIDDLE  ALT=\"24-6=18+non-sophomores%2C%0D%0Aand+they+would+form+%7B%7B%7B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=134596 ALIGN=MIDDLE  ALT=\"134596\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible committees.<BR>\n" );
document.write( "We cannot count that a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > committee.<BR>\n" );
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document.write( "Can a committee of 18 fail to include 2 juniors?<BR>\n" );
document.write( "Yes, while it is impossible to exclude all 7 juniors,<BR>\n" );
document.write( "and form the committee with the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24-7=17 ALIGN=MIDDLE  ALT=\"24-7=17\"   DISABLED_style= \"max-width:90%; height:auto;\" > remaining candidates,<BR>\n" );
document.write( "we can send home 6 juniors,<BR>\n" );
document.write( "and still have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24-6=18 ALIGN=MIDDLE  ALT=\"24-6=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > members for our committee.<BR>\n" );
document.write( "We can do that 7 different ways,<BR>\n" );
document.write( "keeping a different one of the 7 juniors for the committee.<BR>\n" );
document.write( "That way we could make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" > different committees<BR>\n" );
document.write( "that would fail to have the requisite 2 juniors.<BR>\n" );
document.write( "We cannot count those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7 ALIGN=MIDDLE  ALT=\"7\"   DISABLED_style= \"max-width:90%; height:auto;\" > committees, either.<BR>\n" );
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document.write( "Can a committee of 18 fail to include 2 seniors?<BR>\n" );
document.write( "Excluding all 4 seniors,<BR>\n" );
document.write( "we would still have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24-4=20 ALIGN=MIDDLE  ALT=\"24-4=20\"   DISABLED_style= \"max-width:90%; height:auto;\" > students for our committee.<BR>\n" );
document.write( "With those 20 students, we could make<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C20%2C18%29%29=%28matrix%282%2C1%2C20%2C2%29%29=20%2A19%2F%281%2A2%29=190 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2C20%2C18%29%29=%28matrix%282%2C1%2C20%2C2%29%29=20%2A19%2F%281%2A2%29=190\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "different committees that fail the 2-senior requirement,<BR>\n" );
document.write( "because they have no senior.<BR>\n" );
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document.write( "We could also send 3 seniors home,<BR>\n" );
document.write( "and have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24-3=21 ALIGN=MIDDLE  ALT=\"24-3=21\"   DISABLED_style= \"max-width:90%; height:auto;\" > students left to form an 18-member committee.<BR>\n" );
document.write( "that would fail the requirement to have 2 seniors, because it would have only one senior.<BR>\n" );
document.write( "We would have 4 choices as to what senior we would keep,<BR>\n" );
document.write( "and for each of those choices,<BR>\n" );
document.write( "we would still have to pick 17 more students from the 20 non-seniors.<BR>\n" );
document.write( "We would have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C20%2C17%29%29=%28matrix%282%2C1%2C20%2C3%29%29=20%2A19%2A18%2F%281%2A2%2A3%29=1140 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > possible groups of 17 non-seniors,<BR>\n" );
document.write( "that combined with 4 different choices of one senior would make<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A1140=4560 ALIGN=MIDDLE  ALT=\"4%2A1140=4560\"   DISABLED_style= \"max-width:90%; height:auto;\" > committees with only 1 freshman.<BR>\n" );
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document.write( "So, we have to exclude from the total of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=134596 ALIGN=MIDDLE  ALT=\"134596\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible committees<BR>\n" );
document.write( "the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > committee with no sophomore,<BR>\n" );
document.write( "the 7 committees with only 1 junior,<BR>\n" );
document.write( "the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=190 ALIGN=MIDDLE  ALT=\"190\"   DISABLED_style= \"max-width:90%; height:auto;\" > committees with no senior, and<BR>\n" );
document.write( "the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4560 ALIGN=MIDDLE  ALT=\"4560\"   DISABLED_style= \"max-width:90%; height:auto;\" > committees with only 1 freshman.<BR>\n" );
document.write( "Counting them all, that is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B7%2B190%2B4560=4758 ALIGN=MIDDLE  ALT=\"1%2B7%2B190%2B4560=4758\"   DISABLED_style= \"max-width:90%; height:auto;\" > committees that fail the requirement one way or another.<BR>\n" );
document.write( "Could there be one or more committees counted twice in those 4758, because they fail to meet the requirement more than one way?<BR>\n" );
document.write( "The committee that is missing a sophomore includes all the non-sophomores, so it cannot fail by missing any other requirement,<BR>\n" );
document.write( "and the committees that have only 1 junior, must include all non-juniors,, so it cannot fail by missing any other requirement.<BR>\n" );
document.write( "As for the seniors, a committee with only one, and a committee with none cannot be the same, and I think I am counting them properly.<BR>\n" );
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document.write( "Subtracting those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B7%2B190%2B4560=4758 ALIGN=MIDDLE  ALT=\"1%2B7%2B190%2B4560=4758\"   DISABLED_style= \"max-width:90%; height:auto;\" > committees that fail the requirement,<BR>\n" );
document.write( "we are left with<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=134596-4758=highlight%281298388%29 ALIGN=MIDDLE  ALT=\"134596-4758=highlight%281298388%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > committees that meet the requirement.<BR>\n" );
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document.write( "NOTE: The folks at the forum in the artofproblemsolving website would probably instantly answer this question, but they may be too succinct with the explanation, or may feel insulted if it is asked in a forum at a level that considers it too easy a problem;\r<BR>\n" );
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