How Many Unique Ice Cream Sundaes Can You Create?

[thoradicus]

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! ?

 

[LCKurtz]

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?

 

[thoradicus]

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.

 

[LCKurtz]

Yes, that's all there is to it.

 

[pbuk]

For (a) and (b) look again at the number of toppings. You should have spotted this when you tried to calculate 4C6.

 

[thoradicus]

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

 

[Axiomer]

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?

 

[LCKurtz]

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. Altogether too much sugar I guess.