document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171775>Question 1171775</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A pizza parlor has 6 meat options, 8 vegetable options, 3 sauces, and 4 types of cheese.<BR>\n" );
document.write( "Answer the following questions about their menu options (5 points each). If you make<BR>\n" );
document.write( "any assumptions, clearly state what those assumptions are.<BR>\n" );
document.write( "(a) A meats pizza comes mozzarella, the classic sauce, and 4 different types of<BR>\n" );
document.write( "meat. How many different meats pizzas are there?<BR>\n" );
document.write( "(b) The vegan pizza comes with mozzarella, the classic sauce, no meats and any combination of the 8 different vegetables (including the option of having 0 different vegetables). How many different vegetarian pizzas are there?\r<BR>\n" );
document.write( "\n" );
document.write( "(c) The Supremo Pizza comes with your choice of up to 3 meats, up to 4 vegetables,<BR>\n" );
document.write( "one sauce, and up to 2 different cheeses. How many different Supremo pizzas are<BR>\n" );
document.write( "there? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #796700 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=796700\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Part (a)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are 6 meats to choose from, but we only can pick 4.<BR>\n" );
document.write( "There are 6*5*4*3 = 30*12 = 360 different permutations and 360/(4!) = 360/(4*3*2*1) = 360/24 = 15 different combinations. Order doesn't matter when we select the meat, so we go with 15 combinations.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There's 1 choice of cheese (mozzarella), 1 choice of sauce (classic) and 15 combinations of meat. We have 1*1*15 = 15 different meat pizzas possible. You can make a tree diagram to represent all the different combos. A table is also another option.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: 15\r<BR>\n" );
document.write( "\n" );
document.write( "=========================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Part (b)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Consider the set {A,B,C,D,E,F,G,H} which consists of n = 8 items. Each uppercase letter represents a different veggie. The empty set represents picking 0 veggies.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here are a few subsets listed out<BR>\n" );
document.write( "{A,B,C} ... selecting 3 veggies<BR>\n" );
document.write( "{E,F} .... selecting 2 veggies<BR>\n" );
document.write( "{G}  .... selecting 1 veggie<BR>\n" );
document.write( "There are many more subsets possible. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The list of all possible subsets, aka power set, represents all the possible combinations of selecting 0 veggies, 1 veggie, 2 veggies, etc all the way up to 8 veggies.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The number of possible subsets is 2^n = 2^8 = 256. We can check this by listing out every subset, but that will take a while. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There is 1 choice of cheese (mozzarella), 1 choice of sauce (classic), and 256 choices for the veggie combos we discussed above. This leads to 1*1*256 = 256 different vegan pizzas.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: 256\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=========================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Part (c)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For now focus on the meats only.<BR>\n" );
document.write( "We'll use the nCr combination formula here. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The nCr combination formula is<BR>\n" );
document.write( "nCr = (n!)/(r!*(n-r)!)<BR>\n" );
document.write( "where the exclamation marks indicate factorial\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since there are many computations to do, I'll only show the steps for one of them, and let you do the rest (or let you rely on your calculator).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's say we want to pick from a list of 6 meat options and we want to make 3 selections. Order doesn't matter.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We would use n = 6 and r = 3 to get<BR>\n" );
document.write( "nCr = (n!)/(r!*(n-r)!)<BR>\n" );
document.write( "6C3 = (6!)/(3!*(6-3)!)<BR>\n" );
document.write( "6C3 = (6!)/(3!*3!)<BR>\n" );
document.write( "6C3 = (6*5*4*3*2*1)/(3*2*1*3*2*1)<BR>\n" );
document.write( "6C3 = 720/(6*6)<BR>\n" );
document.write( "6C3 = 720/36<BR>\n" );
document.write( "6C3 = 20\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are 20 ways to select exactly 3 meats from a total of 6. Order doesn't matter.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use the nCr formula to calculate when r = 2, r = 1 and r = 0. You should get the following:<BR>\n" );
document.write( "6C2 = 15<BR>\n" );
document.write( "6C1 = 6<BR>\n" );
document.write( "6C0 = 1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "You can use Pascal's triangle to find these values. Look in the row that starts with \"1,6,15,20,...\" and you'll highlight the first four values since they correspond to r=0,r=1,r=2,r=3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "From here we add up the nCr values we found:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(6C3)+(6C2)+(6C1)+(6C0) = (20)+(15)+(6)+(1) = 42\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are 42 ways to select up to 3 meats from 6 meats to choose from.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now onto the veggies. Following the same logic, we get\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(8C4)+(8C3)+(8C2)+(8C1)+(8C0) = (70)+(56)+(28)+(8)+(1) = 163\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are 163 different ways to pick up to 4 veggies from 8 veggies to choose from.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Luckily there's only one sauce, so we can skip over that. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For the cheeses, we have,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(4C2)+(4C1)+(4C0) = (6)+(4)+(1) = 11\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are 11 different ways to select up to 2 cheeses from 4 total cheeses to choose from.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There's a lot going on for part (c), so hopefully everything is making sense so far.  \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To summarize so far we have<ul><li>42 ways to select the meats</li><li>163 ways to select the veggies</li><li>1 way to select the sauce</li><li>11 ways to select the cheeses</li></ul>Multiply out those values:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "42*163*1*11 = 75306\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This represents the number of different Supremo pizzas.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answer: 75306<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );