Combinations and ice cream

In summary, there are 80 possible sundaes using one flavor of ice cream and 3 different toppings, 256 possible sundaes using one flavor of ice cream and from 0 to 6 toppings, and 64 possible combinations of flavors of 3 scoops of ice cream if it is permissible to make all 3 scoops the same flavor.
  • #1
thoradicus
45
0

Homework Statement


To put it simply, there are 4 different flavors of ice cream, and 6 different toppings.

a)how many sundaes are possible using one flavor of ice cream and 3 diff toppings?
b)How many sundaes are possible using one flavour of ice cream and from 0 to 6 toppings?
c)How many different combinations of flavors of 3 scoops of ice cream are possible if it is permissible to make all 3 scoops the same flavor?


Homework Equations


nCr


The Attempt at a Solution


a)4*(4C3) = 80
b)4*(4C0)+4*(4C1) etc up until 6.. I get a 256.
c) This is the part I'm not so sure at...

For each scoop, there are 256 combinations.. so therefore we look at it from the perspective of the different combinations of whether the scoops are of the same flavor or not? like

SSS, SDD, DDD, SSD, same and different

so 256x3! ?
 
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  • #2
thoradicus said:

Homework Statement


To put it simply, there are 4 different flavors of ice cream, and 6 different toppings.

a)how many sundaes are possible using one flavor of ice cream and 3 diff toppings?
b)How many sundaes are possible using one flavour of ice cream and from 0 to 6 toppings?
c)How many different combinations of flavors of 3 scoops of ice cream are possible if it is permissible to make all 3 scoops the same flavor?


Homework Equations


nCr


The Attempt at a Solution


a)4*(4C3) = 80
b)4*(4C0)+4*(4C1) etc up until 6.. I get a 256.
c) This is the part I'm not so sure at...

For each scoop, there are 256 combinations.. so therefore we look at it from the perspective of the different combinations of whether the scoops are of the same flavor or not? like

SSS, SDD, DDD, SSD, same and different

so 256x3! ?

I agree with your first two. For the third, can't you just use the counting principle? How many choices for the first scoop, then for the second, then for the third?
 
  • #3
LCKurtz said:
I agree with your first two. For the third, can't you just use the counting principle? How many choices for the first scoop, then for the second, then for the third?

Oh, so its 4x4x4? since the same flavors are allowed.
 
  • #4
Yes, that's all there is to it.
 
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  • #5
For (a) and (b) look again at the number of toppings. You should have spotted this when you tried to calculate 4C6.
 
  • #6
MrAnchovy said:
For (a) and (b) look again at the number of toppings. You should have spotted this when you tried to calculate 4C6.
hmm not sure why i wrote 4 instead of 6 lol.. but yeah i wrote 6 on my work.. sry for the confusion
 
  • #7
But using the counting principle for c) will distinguish between the same combinations; AAB will be different than ABA etc. The counting principle is used when the order matters, i.e. permutations.

Say your flavours A, B, C and D. You can represent the combination AAA by ***|||, the combination AAB by **|*||, ABC by *|*|*|, BDD by |*||** etc... Can you use this to count the number of combinations of 3 flavours with repetition allowed?
 
  • #8
Axiomer said:
But using the counting principle for c) will distinguish between the same combinations; AAB will be different than ABA etc. The counting principle is used when the order matters, i.e. permutations.

Say your flavours A, B, C and D. You can represent the combination AAA by ***|||, the combination AAB by **|*||, ABC by *|*|*|, BDD by |*||** etc... Can you use this to count the number of combinations of 3 flavours with repetition allowed?

Yes, I think you are correct. I must have been thinking ice cream cones instead of sundaes in my response. :redface: Altogether too much sugar I guess.
 

1. How is ice cream made?

Ice cream is made by combining cream, milk, sugar, and flavorings, and then freezing the mixture while incorporating air. This creates a smooth and creamy texture.

2. What are some popular flavor combinations for ice cream?

Some popular flavor combinations for ice cream include chocolate and peanut butter, vanilla and strawberry, and mint and chocolate chip.

3. Can you make your own combinations of ice cream flavors?

Yes, you can make your own combinations of ice cream flavors. Some popular homemade combinations include coffee and caramel, raspberry and white chocolate, and banana and peanut butter.

4. Are there any unusual combinations of ice cream flavors?

Yes, there are many unusual combinations of ice cream flavors. Some examples include avocado and lime, bacon and maple, and lavender and honey.

5. How does the temperature affect the texture of ice cream?

The temperature plays a crucial role in the texture of ice cream. If the temperature is too high, the ice cream will melt and become soupy. If the temperature is too low, the ice cream will be hard and difficult to scoop. The ideal serving temperature for ice cream is between 6-10 degrees Fahrenheit.

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