Probability Question: 4 layers, 4 flavours

[dacruick]

Okay so here is the problem. I have 4 layers of a cake, and 4 flavours to use. I can make any layer of the cake any flavour, and can use any flavour any amount of times. Also, it does not matter the order of the flavours in the cake. As in 1 1 2 3 is the same as 1 2 1 3 and 3 2 1 1 and so on.

I also believe I have the answer, but do not know why it is the correct answer. I was told that if n represents the flavours, and k represents the layers, the formula goes as follows:

(n + k - 1)! / (n! * (k-1)!). why is that so?

When I originally tried to tackle this problem the method that I took was take 4 x 4 x 4 x 4 to get the total permutations, and then try and subtract or divide out the amount of cakes that can have repeated permutations. Furthermore, I assume that the equation above is doing the same thing, just in a more refined and probabilistically sound manner.

So to sum this up, I would just like someone to explain to me the logic behind the formula. why 7! and so on. Thank you

 

[awkward]

Okay so here is the problem. I have 4 layers of a cake, and 4 flavours to use. I can make any layer of the cake any flavour, and can use any flavour any amount of times. Also, it does not matter the order of the flavours in the cake. As in 1 1 2 3 is the same as 1 2 1 3 and 3 2 1 1 and so on.

I also believe I have the answer, but do not know why it is the correct answer. I was told that if n represents the flavours, and k represents the layers, the formula goes as follows:

(n + k - 1)! / (n! * (k-1)!). why is that so?

When I originally tried to tackle this problem the method that I took was take 4 x 4 x 4 x 4 to get the total permutations, and then try and subtract or divide out the amount of cakes that can have repeated permutations. Furthermore, I assume that the equation above is doing the same thing, just in a more refined and probabilistically sound manner.

So to sum this up, I would just like someone to explain to me the logic behind the formula. why 7! and so on. Thank you

See the section titled "multiset coefficients" here:

http://en.wikipedia.org/wiki/Multiset

 

[Redbelly98]

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Thread moved to:

Homework & Coursework Questions > Precalculus Mathematics​

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I also believe I have the answer, but do not know why it is the correct answer. I was told that if n represents the flavours, and k represents the layers, the formula goes as follows:

(n + k - 1)! / (n! * (k-1)!). why is that so?

When I originally tried to tackle this problem the method that I took was take 4 x 4 x 4 x 4 to get the total permutations, and then try and subtract or divide out the amount of cakes that can have repeated permutations. Furthermore, I assume that the equation above is doing the same thing, just in a more refined and probabilistically sound manner.

So to sum this up, I would just like someone to explain to me the logic behind the formula. why 7! and so on. Thank you

Have you verified that the formula result agrees with your correct, common-sense answer of 4⁴ ?

 

[awkward]

$$4^4$$ is the correct answer if the order of the layers matters, but the problem statement says it does not.

 

[Redbelly98]

Ah, my mistake. awkward is correct.

 

[dacruick]

The thing is, its not a computation, and its not a permutation either. to keep consistent with 'utations', its some sort of mutation between the two. I'm not sure if the derivation of this formula is too advanced for me to really grasp it using logic.

And RedBelly, I am unclear about the rules about which questions go where. I intended this question to be a discussion of the formula, and type of problem that this '4 layers' question represents. As you see I already have the answer, I really just want to know why that is the answer. So if this is the case, do I still have to post it in the homework forum?

 

[tiny-tim]

The thing is, its not a computation, and its not a permutation either. to keep consistent with 'utations', its some sort of mutation between the two. I'm not sure if the derivation of this formula is too advanced for me to really grasp it using logic.

dacruick, this is a combination, as shown quite clearly in the wikipedia article awkward referred you to …

One simple way to prove this involves representing multisets in the following way …

… what do you not understand about that?

(and the fact that you have been given the answer does not make it any the less a homework problem … about half the homework threads ar PF are like that)

 

[dacruick]

If i name the question, what is the different between a combination and a permutation and use that same question to portray my misunderstanding, is it still a homework question. Its not that I have the answer, is the answer that I'm searching for is not a homework question. So is how i name the question the primary factor in where I place it on the forum?