document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=756454>Question 756454</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose you are going to list all the subsets,how many subsets are in {3,6,9,12,15,18 ]?<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 #460208 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=460208\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The set {3,6,9,12,15,18} has 6 elements.<BR>\n" );
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document.write( "NOTE:<BR>\n" );
document.write( "There is an issue with the interpretation of what constitutes a subset.<BR>\n" );
document.write( "I consider the set {3,6,9,12,15,18} to be a sunset of itself, and I consider the empty set to be a subset too. <BR>\n" );
document.write( "So I would have to count all the sets containing 0, 1, 2, 3, 4, 5, and 6 elements from that set. With a different interpretation, the total number of subsets would be less.<BR>\n" );
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document.write( "There is 1 set with 6 elements, and 1 set with 0 elements.<BR>\n" );
document.write( "There are 6 sets with 1 element and 6 sets with 5 elements.<BR>\n" );
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document.write( "IF YOU KNOW AND UNDERSTAND COMBINATION FORMULAS:<BR>\n" );
document.write( "The number of sets of 2, 3, and 4 elements can be calculated as combinations.<BR>\n" );
document.write( "There are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C6%2C2%29%29=6%2A5%2F1%2F2=15 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2C6%2C2%29%29=6%2A5%2F1%2F2=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > sets of 2 elements,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C6%2C3%29%29=6%2A5%2A4%2F1%2F2%2F3=20 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2C6%2C3%29%29=6%2A5%2A4%2F1%2F2%2F3=20\"   DISABLED_style= \"max-width:90%; height:auto;\" > sets of 3 elements, and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C6%2C4%29%29=6%2A5%2A4%2A3%2F1%2F2%2F3%2F4=15 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2C6%2C4%29%29=6%2A5%2A4%2A3%2F1%2F2%2F3%2F4=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > sets of 4 elements.<BR>\n" );
document.write( "(The same combinatorial formulas could have been used to calculate the number of sets of 0, 1, 5, and 6 elements too).<BR>\n" );
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document.write( "WITHOUT USING COMBINATION FORMULAS:<BR>\n" );
document.write( "When assembling sets of 2 elements, you have 6 choices for the first element, and then 5 elements to chose for the second element, for a total of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2A5=30 ALIGN=MIDDLE  ALT=\"6%2A5=30\"   DISABLED_style= \"max-width:90%; height:auto;\" > possible ordered pairs. However, that would count each set twice, with tje elements in different order. For example, you would have counted the ordered pairs (3,6) and (6,3) as 2 ordered pairs, but they represent just one set, the set {3,6}. So there are only <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30%2F2=15 ALIGN=MIDDLE  ALT=\"30%2F2=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > sets of 2 elements.<BR>\n" );
document.write( "The number of sets of 4 elements is also 15, because each set of 2 elements leaves behind a set of 4 elements.<BR>\n" );
document.write( "The number of sets of 3 elements is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20=6%2A5%2A4%2F%283%2A2%29 ALIGN=MIDDLE  ALT=\"20=6%2A5%2A4%2F%283%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > because you could make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%2A5%2A4=120 ALIGN=MIDDLE  ALT=\"6%2A5%2A4=120\"   DISABLED_style= \"max-width:90%; height:auto;\" > ordered triples, but that would count each set of 3 elements <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A2=6 ALIGN=MIDDLE  ALT=\"3%2A2=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > times, because there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A2 ALIGN=MIDDLE  ALT=\"3%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" > different ways to arrange 3 elements (3 choices for which element is first times 2 choices for which one to place second).<BR>\n" );
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document.write( "ADDING UP:<BR>\n" );
document.write( "You can add up the numbers of sets with 0, 1, 2, 3, 4, 5, and 6 elements the simple way:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B6%2B15%2B20%2B15%2B6%2B1=highlight%2864%29 ALIGN=MIDDLE  ALT=\"1%2B6%2B15%2B20%2B15%2B6%2B1=highlight%2864%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The fancy way to add would be to add the combinations as a power of a binomial<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C6%2C0%29%29%2B%28matrix%282%2C1%2C6%2C1%29%29%2B%28matrix%282%2C1%2C6%2C2%29%29%2B%28matrix%282%2C1%2C6%2C3%29%29%2B%28matrix%282%2C1%2C6%2C4%29%29%2B%28matrix%282%2C1%2C6%2C5%29%29%2B%28matrix%282%2C1%2C6%2C6%29%29=%281%2B1%29%5E6=2%5E6=highlight%2864%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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