document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1029837>Question 1029837</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Could someone please help me with this problem? I think I may be on the right track, but am not really sure!<BR>\n" );
document.write( "\"From a squad of 12 cheerleaders, 10 will assemble themselves into a 4-level pyramid.<BR>\n" );
document.write( "a) how many different combinations of cheerleaders can be used to build the pyramid? **I think it's 12/10 or 12 C 10 (sub 12, C, sub 10/don't know how to get it on here correctly, sorry!) which equals 66, but I don't understand how to work it out.<BR>\n" );
document.write( "b) the coach decides that Alexis must be at the top of the pyramid, and that Rachel, Jen, Britt, and Nicole will form the base.  How many different combinations of 10 cheerleaders can now be chosen to form the pyramid?<BR>\n" );
document.write( "**I think it's 7/5 and answer is 21, but again I don't know for sure or how to actually arrive at that answer<BR>\n" );
document.write( "c) Suppose alexis has an injury and can't participate. the coach replaces Alexis with Rebecca. Now, how many different combinations of 10 cheerleaders can the coach select?<BR>\n" );
document.write( "*I think it's 6/5 which equals 6, same on this one, not sure how to work out. Or, if my answers are even correct??<BR>\n" );
document.write( "Any help would be appreciated! Thanks! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #644902 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=644902\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> There are originally <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > cheerleaders that can participate.<BR>\n" );
document.write( "Of those <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > will form a pyramid.<BR>\n" );
document.write( "The other <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12-10=2 ALIGN=MIDDLE  ALT=\"12-10=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > will just watch.<BR>\n" );
document.write( "As the problem's wording suggests, it is a question of combinations (where order/position in the pyramid does not matter).<BR>\n" );
document.write( "It is not a question of permutations.<BR>\n" );
document.write( " <BR>\n" );
document.write( "a) The number of different combinations (subsets) 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;\" > cheerleaders<BR>\n" );
document.write( "that can be made from that set of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > available cheerleaders can be calculated by a formula or just reasoned through.<BR>\n" );
document.write( "The written and applied formula may be required by the teacher.<BR>\n" );
document.write( "Reasoning is all that is needed to get to the solution and/or the formula.<BR>\n" );
document.write( "The formula for the number of combinations (subsets) 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;\" > objects is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2Cn%2Cr%29%29=n%21%2F%28%28n-r%29%21r%21%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2Cn%2Cr%29%29=n%21%2F%28%28n-r%29%21r%21%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "In this case,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C12%2C10%29%29=12%21%2F%28%2812-10%29%2110%21%29=12%21%2F%282%2110%21%29=12%2A11%2A10%2A9%2A8%2A7%2A6%2A5%2A4%2A3%2A2%2A1%2F%28%282%2A1%29%2810%2A9%2A8%2A7%2A6%2A5%2A4%2A3%2A2%2A1%29%29=12%2A11%2F2=66 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You could look at it from the other side.<BR>\n" );
document.write( "The number of different possible combinations 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;\" > cheerleaders to be used to make the pyramid is the same as<BR>\n" );
document.write( "the number of different possible combinations of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > disappointed cheerleaders can be chosen to not participate in the pyramid.<BR>\n" );
document.write( "Reasoning and calculating the answer that way is easier.<BR>\n" );
document.write( "The coach could look at the cheerleaders and decide to leave out the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > of them who have missed the most practices.<BR>\n" );
document.write( "There would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12 ALIGN=MIDDLE  ALT=\"12\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways to make his first choice of cheerleader to exclude,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11 ALIGN=MIDDLE  ALT=\"11\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways to make the second choice.<BR>\n" );
document.write( "So the coach's thinking could be made in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2A11 ALIGN=MIDDLE  ALT=\"12%2A11\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways when order of choices matter.<BR>\n" );
document.write( "But since the same set of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > excluded cheerleaders could be picked <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > different ways,<BR>\n" );
document.write( "the number of different possible sets of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > excluded cheerleaders is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2A11%2F2=66 ALIGN=MIDDLE  ALT=\"12%2A11%2F2=66\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Applying the formula, we would calculate it as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C12%2C2%29%29=12%21%2F%282%21%2812-2%29%21%29=12%21%2F%282%2110%21%29=66 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2C12%2C2%29%29=12%21%2F%282%21%2812-2%29%21%29=12%21%2F%282%2110%21%29=66\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "b) Once the coach decide on the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > cheerleaders that will be the top and the base of the pyramid,<BR>\n" );
document.write( "it is a question of how many different sets of other cheerleaders can the coach choose to form the other two layers to complete the pyramid.<BR>\n" );
document.write( "There are still <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12-5=7 ALIGN=MIDDLE  ALT=\"12-5=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > cheerleaders to choose from for the other <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10-5=5 ALIGN=MIDDLE  ALT=\"10-5=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > pyramid positions.<BR>\n" );
document.write( "The number of possible choices is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C7%2C5%29%29=7%21%2F%28%287-5%29%215%21%29=7%21%2F%282%215%21%29=21 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2C7%2C5%29%29=7%21%2F%28%287-5%29%215%21%29=7%21%2F%282%215%21%29=21\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You can also reason there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2A6%2F2=21 ALIGN=MIDDLE  ALT=\"7%2A6%2F2=21\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways to pick the two cheerleaders who will not participate.<BR>\n" );
document.write( " <BR>\n" );
document.write( "c)With Alexis injured, there are only <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12-1=11 ALIGN=MIDDLE  ALT=\"12-1=11\"   DISABLED_style= \"max-width:90%; height:auto;\" > available cheerleaders.<BR>\n" );
document.write( "Other than that, it is like the situation in part b:<BR>\n" );
document.write( "after the coach picks <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > cheerleaders for specific positions in the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > person pyramid,<BR>\n" );
document.write( "there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10-5=5 ALIGN=MIDDLE  ALT=\"10-5=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > more cheerleaders to be chosen<BR>\n" );
document.write( "out of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11-5=6 ALIGN=MIDDLE  ALT=\"11-5=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > remaining available cheerleaders.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C6%2C5%29%29=6%21%2F%28%286-5%29%215%21%29=6%21%2F%281%215%21%29=61 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2C6%2C5%29%29=6%21%2F%28%286-5%29%215%21%29=6%21%2F%281%215%21%29=61\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You can also reason there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways to pick 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;\" > of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6 ALIGN=MIDDLE  ALT=\"6\"   DISABLED_style= \"max-width:90%; height:auto;\" > cheerleaders who will not participate.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "THE REASONING FOR THE FORMULA:<BR>\n" );
document.write( "The reasoning is that when you make the list of the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r ALIGN=MIDDLE  ALT=\"r\"   DISABLED_style= \"max-width:90%; height:auto;\" > objects,<BR>\n" );
document.write( "there are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > possibilities for the first object to be listed,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-1 ALIGN=MIDDLE  ALT=\"n-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > possibilities for the second one, and so on,<BR>\n" );
document.write( "so the list is being built <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2A%28n-1%29%2A%22...%22 ALIGN=MIDDLE  ALT=\"n%2A%28n-1%29%2A%22...%22\"   DISABLED_style= \"max-width:90%; height:auto;\" > ways.<BR>\n" );
document.write( "The number of ways is a product of consecutive factors counting down from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "After the last item is chosen, there are <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;\" > objects not chosen,<BR>\n" );
document.write( "so the final product does not include <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n-r%29%2A%28n-r-1%29%2A%22...%22%2A3%2A2%2A1=%28n-r%29%21 ALIGN=MIDDLE  ALT=\"%28n-r%29%2A%28n-r-1%29%2A%22...%22%2A3%2A2%2A1=%28n-r%29%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "but all the other factors in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%21=n%2A%28n-1%29%2A%22...%22%2A3%2A2%2A1 ALIGN=MIDDLE  ALT=\"n%21=n%2A%28n-1%29%2A%22...%22%2A3%2A2%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > are included.<BR>\n" );
document.write( "The final product is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-%28n-1%29%2A%22...%22%2A%28n-r%2B1%29 ALIGN=MIDDLE  ALT=\"n-%28n-1%29%2A%22...%22%2A%28n-r%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2A%28n-1%29%2A%22...%22%2A%28n-r%2B1%29%2A%28n-r%29%2A%28n-r-1%29%2A%22...%22%2A3%2A2%2A1%2F%28%28n-r%29%2A%28n-r-1%29%2A%22...%22%2A3%2A2%2A1%29=n%21%2F%28n-r%29%21 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , and that is the number of ways the list could be built.<BR>\n" );
document.write( "However, a list 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;\" > objects can be written <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%21=r%2A%28r-1%29%2A%22...%22%2A3%2A2%2A1 ALIGN=MIDDLE  ALT=\"r%21=r%2A%28r-1%29%2A%22...%22%2A3%2A2%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > different ways,<BR>\n" );
document.write( "because there are <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 could be first, and for each case, there would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r-1 ALIGN=MIDDLE  ALT=\"r-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > items that could be second, and so on.<BR>\n" );
document.write( "Since the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%21%2F%28n-r%29%21 ALIGN=MIDDLE  ALT=\"n%21%2F%28n-r%29%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > lists include <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%21 ALIGN=MIDDLE  ALT=\"r%21\"   DISABLED_style= \"max-width:90%; height:auto;\" > repeats of each combination/subset 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;\" > objects,<BR>\n" );
document.write( "the number of combinations 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;\" > objects that can be made from a set 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;\" > objects is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2Cn%2Cr%29%29=%28%28n%21%2F%28n-r%29%21%29%29%2Fr%21=n%21%2F%28n-r%29%21r%21%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );