document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1161853>Question 1161853</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Formula for question (a): nPr=n!/r!(n-r)! Assume a class has 24 members.<BR>\n" );
document.write( "(a)How many 4-person committees can be formed out of a class of 15?\r<BR>\n" );
document.write( "\n" );
document.write( "Formula for (b): The number of combinations of n objects taken r at a time is given by the formula nCr=n!/r!(n-r_!<BR>\n" );
document.write( "(b)How many committees of 10 people can be chosen?\r<BR>\n" );
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document.write( "  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #785534 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=785534\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You have a wrong formula, and formulas are not good for brain health.<BR>\n" );
document.write( "For a student, understanding and reasoning is best for brain health.<BR>\n" );
document.write( "You cannot raise wise students feeding them formulas.<BR>\n" );
document.write( "(You can figure out by yourself how to feed babies for best brain health).<BR>\n" );
document.write( " <BR>\n" );
document.write( "However, to communicate with computers, calculators, and math teachers<BR>\n" );
document.write( "we need a little vocabulary/symbols. (More of that later).<BR>\n" );
document.write( "The symbol <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%21 ALIGN=MIDDLE  ALT=\"4%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A3%2A2%2A1=24 ALIGN=MIDDLE  ALT=\"4%2A3%2A2%2A1=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "a product with the first 4 counting numbers as factors.<BR>\n" );
document.write( "We call it \"factorial of 4\" or \"4 factorial\".<BR>\n" );
document.write( "My calculator has \"x!\" as a function,<BR>\n" );
document.write( "but factorial is used for positive integers,<BR>\n" );
document.write( "although we extend it beyond <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%21=1 ALIGN=MIDDLE  ALT=\"1%21=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%21=1 ALIGN=MIDDLE  ALT=\"0%21=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "(a) I understand that when we pick <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > students for a committee out of a class of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15 ALIGN=MIDDLE  ALT=\"15\"   DISABLED_style= \"max-width:90%; height:auto;\" > students,<BR>\n" );
document.write( "we just need a group/set of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > students.<BR>\n" );
document.write( "We can list them in any order and it is the same committee<BR>\n" );
document.write( "with all students equally important for now.<BR>\n" );
document.write( "They can distribute their roles and responsibilities among themselves later.<BR>\n" );
document.write( "We could instead make an ordered list,<BR>\n" );
document.write( "where the first named is the president of the committee,<BR>\n" );
document.write( "the second one is the secretary/note-taker,<BR>\n" );
document.write( "the third is the treasurer,<BR>\n" );
document.write( "and the last one brings the donuts to the meetings.<BR>\n" );
document.write( "In that case the same 4 people mentioned in different order make a different list.<BR>\n" );
document.write( "When we make lists of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > people out of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15 ALIGN=MIDDLE  ALT=\"15\"   DISABLED_style= \"max-width:90%; height:auto;\" > available to choose from,<BR>\n" );
document.write( "we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15 ALIGN=MIDDLE  ALT=\"15\"   DISABLED_style= \"max-width:90%; height:auto;\" > choices for the first name in the list.<BR>\n" );
document.write( "Afterwards, for each of those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15 ALIGN=MIDDLE  ALT=\"15\"   DISABLED_style= \"max-width:90%; height:auto;\" > choices,<BR>\n" );
document.write( "we will have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14 ALIGN=MIDDLE  ALT=\"14\"   DISABLED_style= \"max-width:90%; height:auto;\" > remaining choices for a second name (the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14 ALIGN=MIDDLE  ALT=\"14\"   DISABLED_style= \"max-width:90%; height:auto;\" > not yet chosen students).<BR>\n" );
document.write( "Then, we will have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13 ALIGN=MIDDLE  ALT=\"13\"   DISABLED_style= \"max-width:90%; height:auto;\" > choices for the third name,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > choices for the fourth.<BR>\n" );
document.write( "All in all, we could make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%2A14%2A13%2A12=32760 ALIGN=MIDDLE  ALT=\"15%2A14%2A13%2A12=32760\"   DISABLED_style= \"max-width:90%; height:auto;\" > lists.<BR>\n" );
document.write( "That is not <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32760 ALIGN=MIDDLE  ALT=\"32760\"   DISABLED_style= \"max-width:90%; height:auto;\" > different sets of students,<BR>\n" );
document.write( "because each set of 4 students can be listed <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2A3%2A2%2A1=24 ALIGN=MIDDLE  ALT=\"4%2A3%2A2%2A1=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > different ways Different orders).<BR>\n" );
document.write( "We will find each different set of 4 students repeated as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=24 ALIGN=MIDDLE  ALT=\"24\"   DISABLED_style= \"max-width:90%; height:auto;\" > times in those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32760 ALIGN=MIDDLE  ALT=\"32760\"   DISABLED_style= \"max-width:90%; height:auto;\" > lists.<BR>\n" );
document.write( "Those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=32760 ALIGN=MIDDLE  ALT=\"32760\"   DISABLED_style= \"max-width:90%; height:auto;\" > lists represent only<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2815%2A14%2A13%2A12%29%2F%284%2A3%2A2%2A1%29=32760%2F24=highlight%281365%29 ALIGN=MIDDLE  ALT=\"%2815%2A14%2A13%2A12%29%2F%284%2A3%2A2%2A1%29=32760%2F24=highlight%281365%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > different committees.<BR>\n" );
document.write( " <BR>\n" );
document.write( "(b) For a list of 10 students out of a class of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15 ALIGN=MIDDLE  ALT=\"15\"   DISABLED_style= \"max-width:90%; height:auto;\" > there are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%2A14%2A13%2A%22....%22%2A8%2A7%2A6 ALIGN=MIDDLE  ALT=\"15%2A14%2A13%2A%22....%22%2A8%2A7%2A6\"   DISABLED_style= \"max-width:90%; height:auto;\" > different possibilities<BR>\n" );
document.write( "That is a product of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors that are the numbers from 1 to 15, except the first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15-10=5 ALIGN=MIDDLE  ALT=\"15-10=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > numbers.<BR>\n" );
document.write( "We would tell a computer or a calculator to calculate that as computer could see that as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2815%2A14%2A13%2A%22....%22%2A8%2A7%2A6%2A5%2A4%2A3%2A2%2A10%29%2F%285%2A4%2A3%2A2%2A1%29=15%21%2F%2815-10%29%21=15%21%2F5%21 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Each set of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%5D ALIGN=MIDDLE  ALT=\"1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > students can be written in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%21 ALIGN=MIDDLE  ALT=\"10%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > different orders as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%21 ALIGN=MIDDLE  ALT=\"10%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > different lists,<BR>\n" );
document.write( "so the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%21%2F%2815-10%29%21=15%21%2F5%21 ALIGN=MIDDLE  ALT=\"15%21%2F%2815-10%29%21=15%21%2F5%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > different lists represent<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15%21%2F%2815-10%29%21%2F10%21=15%21%2F5%21%2F10%21=highlight%28273%29 ALIGN=MIDDLE  ALT=\"15%21%2F%2815-10%29%21%2F10%21=15%21%2F5%21%2F10%21=highlight%28273%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "MORE symbols and vocabulary:<BR>\n" );
document.write( "My (cheap) scientific calculator has keys for the functions <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=nPr ALIGN=MIDDLE  ALT=\"nPr\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=nCr ALIGN=MIDDLE  ALT=\"nCr\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=nPr ALIGN=MIDDLE  ALT=\"nPr\"   DISABLED_style= \"max-width:90%; height:auto;\" > (with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%3E=r ALIGN=MIDDLE  ALT=\"n%3E=r\"   DISABLED_style= \"max-width:90%; height:auto;\" > , of course) represents<BR>\n" );
document.write( "the number of different permutations (lists) of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > items that can be made<BR>\n" );
document.write( "out of a total of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > different items.<BR>\n" );
document.write( "It is called permutations of n taken r at a time because the same r items in different order make a different list/permutation.<BR>\n" );
document.write( "The formula to calculate that number is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=nPr=n%21%2F%28n-r%29%21 ALIGN=MIDDLE  ALT=\"nPr=n%21%2F%28n-r%29%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > , where<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%21=n%2A%28n-1%29%2A%28n-2%29%2A%22...%22%2A3%2A2%2A1 ALIGN=MIDDLE  ALT=\"n%21=n%2A%28n-1%29%2A%28n-2%29%2A%22...%22%2A3%2A2%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the product of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > consecutive integers from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n-r%29%21=n%2A%28n-r-1%29%2A%28n-r-2%29%2A%22...%22%2A3%2A2%2A1 ALIGN=MIDDLE  ALT=\"%28n-r%29%21=n%2A%28n-r-1%29%2A%28n-r-2%29%2A%22...%22%2A3%2A2%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the product of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-r ALIGN=MIDDLE  ALT=\"n-r\"   DISABLED_style= \"max-width:90%; height:auto;\" > consecutive integers from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-r ALIGN=MIDDLE  ALT=\"n-r\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "(except that we define <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%21=1 ALIGN=MIDDLE  ALT=\"0%21=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > for consistency).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=nPn=n%21%2F%28n-n%29%21=n%21%2F0%21=n%21%2F1=n%21 ALIGN=MIDDLE  ALT=\"nPn=n%21%2F%28n-n%29%21=n%21%2F0%21=n%21%2F1=n%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents the number of list permutations of n items<BR>\n" );
document.write( "that we can make using all n items at a time (for each list).<BR>\n" );
document.write( "We just call it permutations of n, but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%21 ALIGN=MIDDLE  ALT=\"n%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > is shorter and more direct.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=nCr=nPr%2Fn%21 ALIGN=MIDDLE  ALT=\"nCr=nPr%2Fn%21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22n%21%2F%28n-r%29%21%22%2Fr%21 ALIGN=MIDDLE  ALT=\"%22n%21%2F%28n-r%29%21%22%2Fr%21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%22=%22 ALIGN=MIDDLE  ALT=\"%22=%22\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%21%2F%28n-r%29%21%2Fr%21 ALIGN=MIDDLE  ALT=\"n%21%2F%28n-r%29%21%2Fr%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents the number of different sets/combinations you can make<BR>\n" );
document.write( "by picking  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > items out of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > available items.<BR>\n" );
document.write( "We use \"C\" in that symbol for combinations,<BR>\n" );
document.write( "meaning that we consider {lettuce,tomato,pickle} the same<BR>\n" );
document.write( "set/combination of 3 burger toppings as {tomato,pickle,lettuce} . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );