How to distinguish between N and K? (Collections W/ Repetition)

[tryingtoexcelatmath]

Suppose you want to distribute 15 ham sandwiches to eight people, with no restriction on how many each person gets.

How many ways are there of choosing how many sandwiches each person gets?

In the problem above, what is the N and what is the K?

I can't figure out what is N and what is K. 15 sandwiches and 8 people. Which one is the N and which one is the K?

This is for the formula N-1+K choose K (or choose N-1)

And..

You roll seven identical six-sided dice simultaneously.
How many different outcomes are possible? (For instance, one possible outcome is three 2 s, one 6 , one 5 , and two 1 s.)

For the problem above what is the N and what is the K?

I can't seem to figure out which one is N (the 7 dice or the 6 sides) and which is the K (the 7 dice or the 6 sides)

This is for the formula N-1+K choose K (or choose N-1)

Is there some method to figure out what is N and what is K for all types of problems?

 

[HallsofIvy]

Suppose you want to distribute 15 ham sandwiches to eight people, with no restriction on how many each person gets.

How many ways are there of choosing how many sandwiches each person gets?

In the problem above, what is the N and what is the K?

I can't figure out what is N and what is K. 15 sandwiches and 8 people. Which one is the N and which one is the K?

This is for the formula N-1+K choose K (or choose N-1)

And..

You roll seven identical six-sided dice simultaneously.
How many different outcomes are possible? (For instance, one possible outcome is three 2 s, one 6 , one 5 , and two 1 s.)

For the problem above what is the N and what is the K?

Do you not recognize that this makes no sense at all? There is no "K" or "N" in what you say above! I presume you are referring to some formula that involves the letters N and K but you haven't told us what formula you are referring to.

IF we counted order as different, for example counting "1, 3, 6, 5, 4, 3, 2" as different from "3, 6, 2, 4, 3, 1, 5" then there would be \(\displaystyle 6^7\) possible outcomes.
But it appears you are not, that both of those would be the same outcome "one 1, one 2, two 3s, one 4, one 5, and one 6". So the number of outcomes is smaller than that, \(\displaystyle 6^7\) divided by the number "equivalent" outcomes.

I can't seem to figure out which one is N (the 7 dice or the 6 sides) and which is the K (the 7 dice or the 6 sides)

This is for the formula N-1+K choose K (or choose N-1)

Is there some method to figure out what is N and what is K for all types of problems?

 

[tryingtoexcelatmath]

The formula that I am referring to is here:

This is for the formula N-1+K choose K (or choose N-1)

Sorry for the confusion! I am learning collections w/ repetitions so the formula is:

n-1+k choose k

Suppose you want to distribute 15 ham sandwiches to eight people, with no restriction on how many each person gets.

How many ways are there of choosing how many sandwiches each person gets?

So if K = 15 sandwiches
N = 8 people (thus n-1 = 7 people)
N-1+K = 22

So 22 choose 15 will give you the answer that was given in the course.

My question is how do we know if 15 ham sandwiches is N or K, and if 8 people is N or K?

Similarly:
You roll seven identical six-sided dice simultaneously.
How many different outcomes are possible?

If we apply the (N-1+K choose K) formula, what is N and what is K? is N the sides of the dice (6)? or is N the the number of dice (7)?

The course says the answer is:

N = sides of the dice which is 6, so N-1 = 5
K = 7 the number of dice

And using this information you can solve for (N-1+K choose K)

But why is N = the sides of the dice instead of N = the number of dice?

 

[pka]

Suppose you want to distribute 15 ham sandwiches to eight people, with no restriction on how many each person gets.
How many ways are there of choosing how many sandwiches each person gets?
In the problem above, what is the N and what is the K?
I can't figure out what is N and what is K. 15 sandwiches and 8 people. Which one is the N and which one is the K?
This is for the formula N-1+K choose K (or choose N-1)

The formula to which you refer is \(\dbinom{N+K-1}{N}\), the number of ways to place \(N\) identical objects into \(K\) distinct cells.
In this question we assume the sandwiches are identical and people are distinct: \(\dbinom{15+8-1}{15}\)

 

[tryingtoexcelatmath]

The formula to which you refer is \(\dbinom{N+K-1}{N}\), the number of ways to place \(N\) identical objects into \(K\) distinct cells.
In this question we assume the sandwiches are identical and people are distinct: \(\dbinom{15+8-1}{15}\)

N = identical
K = distinct

Thanks! I think that does help me understand it better.

So for the dice, we have 7 distinct dice which is K, and 6 identical sides? which is N?

 

[tryingtoexcelatmath]

What about in this example?

By a rack in the game of Scrabble, we'll mean a collection of seven letters drawn from an alphabet comprised of the usual 26 English letters. The order doesn't matter, and you can have more than one of a given letter.
(For the purposes of this problem, we'll say there are seven of each letter available and no blank tiles.)
How many different racks are possible?

So we have 7 tiles _ _ _ _ _ _ _

and each tile have 26 english letter possibilities

What is N and what is K?

I just want to check the answer against your logic of N = identical and K = distinct

In the answer they gave N = 7 so N-1 = 6

And K = 26

So.. the 7 tiles are identical? and the 26 letters are distinct?

To be honest at first when I tried to apply the identical / distinct logic, I got it the other way.

I thought it was N = 26 identical objects placed into K = 7 distinct cells?

 

[Dr.Peterson]

The issue is not, what is the formula; yes, you showed that. The issue is, what is it a formula for? How did your source define n and k? That's what you need to know in order to apply the formula.

Please answer that question (though pka has done so separately). What is your source? What did it say? By examining that we can help you learn to apply that and other formulas.

I personally prefer not to memorize formulas; I solve this kind of problem using the "stars and bars" method, which requires not memorization of what it applies to; it amounts to deriving the formula when you need it, and takes no longer.

There are other issues that factor into deciding which formula to use, and how. One is, is everyone expected to get at least one sandwich?

 

[Dr.Peterson]

What about in this example?

By a rack in the game of Scrabble, we'll mean a collection of seven letters drawn from an alphabet comprised of the usual 26 English letters. The order doesn't matter, and you can have more than one of a given letter.
(For the purposes of this problem, we'll say there are seven of each letter available and no blank tiles.)
How many different racks are possible?

So we have 7 tiles _ _ _ _ _ _ _

and each tile have 26 english letter possibilities

What is N and what is K?

I just want to check the answer against your logic of N = identical and K = distinct

In the answer they gave N = 7 so N-1 = 6

And K = 26

So.. the 7 tiles are identical? and the 26 letters are distinct?

To be honest at first when I tried to apply the identical / distinct logic, I got it the other way.

I thought it was N = 26 identical objects placed into K = 7 distinct cells?

The main issue in defining N and K in your formula (which I expect your source to have said) is not that there are some identical things and some distinct things, but that all of some [identical, unordered] things are each associated with some or all of some other [distinct] things, so that a total number N is split up into a sum of K numbers representing how many of the former are "in" each of the latter.

Are all of the letters used? Or are all of the slots in the rack used? Then, are all letters treated as identical (interchangeable)? Are the tiles treated as distinct (so that it matters which has what letter)? Is the problem equivalent to putting every letter into a place, or putting every tile into a place on the alphabet? Those are among the questions to ask.

 

[pka]

By a rack in the game of Scrabble, we'll mean a collection of seven letters drawn from an alphabet comprised of the usual 26 English letters. The order doesn't matter, and you can have more than one of a given letter.
(For the purposes of this problem, we'll say there are seven of each letter available and no blank tiles.)
How many different racks are possible?

The act of choosing seven times is in fact making seven identical actions (choices) from twenty-six different letters. Thus \(\dbinom{7+26-1}{7}\).
An ice-cream shop has twenty-six flavors from which one can choose three for a banana split. How many different splits are there?
The acts of choosing are identical so \(\dbinom{3+26-1}{3}\).

 

[tryingtoexcelatmath]

The main issue in defining N and K in your formula (which I expect your source to have said) is not that there are some identical things and some distinct things, but that all of some [identical, unordered] things are each associated with some or all of some other [distinct] things, so that a total number N is split up into a sum of K numbers representing how many of the former are "in" each of the latter.

Are all of the letters used? Or are all of the slots in the rack used? Then, are all letters treated as identical (interchangeable)? Are the tiles treated as distinct (so that it matters which has what letter)? Is the problem equivalent to putting every letter into a place, or putting every tile into a place on the alphabet? Those are among the questions to ask.

Hi Dr Peterson: I see ok thanks.

For reference I am taking a course through EDX:

Authentication | edX

courses.edx.org

Lesson 2.5 is going through collection w/ repetition

The act of choosing seven times is in fact making seven identical actions (choices) from twenty-six different letters. Thus \(\dbinom{7+26-1}{7}\).
An ice-cream shop has twenty-six flavors from which one can choose three for a banana split. How many different splits are there?
The acts of choosing are identical so \(\dbinom{3+26-1}{3}\).

I see, I think I follow your logic regarding making 7 identical actions from 26 different letters.

Basically K is whatever goes into N.

If it was the ice cream shop and we want to know how many ice cream flavors can go into a banana split, we take ice cream as K and the split as N.