Question 1171775: A pizza parlor has 6 meat options, 8 vegetable options, 3 sauces, and 4 types of cheese.
Answer the following questions about their menu options (5 points each). If you make
any assumptions, clearly state what those assumptions are.
(a) A meats pizza comes mozzarella, the classic sauce, and 4 different types of
meat. How many different meats pizzas are there?
(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?
(c) The Supremo Pizza comes with your choice of up to 3 meats, up to 4 vegetables,
one sauce, and up to 2 different cheeses. How many different Supremo pizzas are
there?
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Part (a)
There are 6 meats to choose from, but we only can pick 4.
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.
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.
Answer: 15
=========================================
Part (b)
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.
Here are a few subsets listed out
{A,B,C} ... selecting 3 veggies
{E,F} .... selecting 2 veggies
{G} .... selecting 1 veggie
There are many more subsets possible.
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.
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.
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.
Answer: 256
=========================================
Part (c)
For now focus on the meats only.
We'll use the nCr combination formula here.
The nCr combination formula is
nCr = (n!)/(r!*(n-r)!)
where the exclamation marks indicate factorial
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).
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.
We would use n = 6 and r = 3 to get
nCr = (n!)/(r!*(n-r)!)
6C3 = (6!)/(3!*(6-3)!)
6C3 = (6!)/(3!*3!)
6C3 = (6*5*4*3*2*1)/(3*2*1*3*2*1)
6C3 = 720/(6*6)
6C3 = 720/36
6C3 = 20
There are 20 ways to select exactly 3 meats from a total of 6. Order doesn't matter.
----------------
Use the nCr formula to calculate when r = 2, r = 1 and r = 0. You should get the following:
6C2 = 15
6C1 = 6
6C0 = 1
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
From here we add up the nCr values we found:
(6C3)+(6C2)+(6C1)+(6C0) = (20)+(15)+(6)+(1) = 42
There are 42 ways to select up to 3 meats from 6 meats to choose from.
----------------
Now onto the veggies. Following the same logic, we get
(8C4)+(8C3)+(8C2)+(8C1)+(8C0) = (70)+(56)+(28)+(8)+(1) = 163
There are 163 different ways to pick up to 4 veggies from 8 veggies to choose from.
----------------
Luckily there's only one sauce, so we can skip over that.
For the cheeses, we have,
(4C2)+(4C1)+(4C0) = (6)+(4)+(1) = 11
There are 11 different ways to select up to 2 cheeses from 4 total cheeses to choose from.
----------------
There's a lot going on for part (c), so hopefully everything is making sense so far.
To summarize so far we have- 42 ways to select the meats
- 163 ways to select the veggies
- 1 way to select the sauce
- 11 ways to select the cheeses
Multiply out those values:
42*163*1*11 = 75306
This represents the number of different Supremo pizzas.
----------------
Answer: 75306
|
|
|
