document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1133505>Question 1133505</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A pizza parlor has a choice of <BR>\n" );
document.write( "12\r<BR>\n" );
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document.write( " toppings for its pizzas. From these <BR>\n" );
document.write( "12\r<BR>\n" );
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document.write( " toppings, how many different <BR>\n" );
document.write( "6\r<BR>\n" );
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document.write( "-topping pizzas can be ordered? Assume that the order in which the toppings are listed does not matter.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #750743 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=750743\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Basic combinatorics principles -- permutations and combinations.<br><BR>\n" );
document.write( "If order DID matter, you would have 12 choices for the first topping, then 11 for the second, ..., and 7 for the sixth.  The total number of ORDERED sets of toppings (permutations) would be<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2A11%2A10%2A9%2A8%2A7+=+665280 ALIGN=MIDDLE  ALT=\"12%2A11%2A10%2A9%2A8%2A7+=+665280\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That number is \"12P6\", the number of ordered ways (permutations) to choose 6 of 12 things.<br><BR>\n" );
document.write( "In your problem, the order does NOT matter.  For that we have the number of combinations of 6 of 12 things, \"12C6\".  To calculate that number, we take the number 12P6, the number of ORDERED ways we could choose 6 of the 12 toppings, and divide it by the number of different ORDERS in which we could have chosen those 6 toppings.<br><BR>\n" );
document.write( "Similar to the reasoning above, we would have 6 choices for the first one we chose, 5 choices for the second, ..., and 1 choice for the sixth.  So the number of different ways we could have chosen any particular 6 toppings is 6*5*4*3*2*1 = 720.<br><BR>\n" );
document.write( "Then the number of COMBINATIONS (order doesn't matter) of 6 of 12 toppings is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=665280%2F720+=+924 ALIGN=MIDDLE  ALT=\"665280%2F720+=+924\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "If you don't have a calculator to do the calculations for you, the calculation to do by hand is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2812%2A11%2A10%2A9%2A8%2A7%29%2F%286%2A5%2A4%2A3%2A2%2A1%29 ALIGN=MIDDLE  ALT=\"%2812%2A11%2A10%2A9%2A8%2A7%29%2F%286%2A5%2A4%2A3%2A2%2A1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "There will always be lots of common factors with which you can simplify the fraction; and if you do the simplification correctly the answer will always be a whole number.<br><BR>\n" );
document.write( "For example, in this calculation 5*2=10 and 4*3=12, so a partially simplified fraction is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2811%2A9%2A8%2A7%29%2F6 ALIGN=MIDDLE  ALT=\"%2811%2A9%2A8%2A7%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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