Cheese shop combination question

[RoboNerd]

Homework Statement

Question:

A cheese shop carries a large stock of 34 kinds of cheese. By the end of the day 48 cheese sales have been made and the items sold must be restocked. How many different restocking orders are possible?

Homework Equations

Combination and permutation equations

The Attempt at a Solution

[/B]
Hi Everyone! I have the above question and have no idea on how to get started.

Well I do know that each of the sales must involve buying at least one type of cheese, so this sounds to me like a stars-and-bars combination type problem. However the fact that we are looking for the different combinations of 48 cheese sales instead of one has me stumped on what approach I should take.

So I need to figure out how many different types of orders have taken place, but I have no idea on how to start. Could anyone please help guide me in the right direction?

Thanks in advance.

 

[Mark44]

Homework Statement

Question:

A cheese shop carries a large stock of 34 kinds of cheese. By the end of the day 48 cheese sales have been made and the items sold must be restocked. How many different restocking orders are possible?

Homework Equations

Combination and permutation equations

Such as?

The Attempt at a Solution

[/B]
Hi Everyone! I have the above question and have no idea on how to get started.

Well I do know that each of the sales must involve buying at least one type of cheese, so this sounds to me like a stars-and-bars combination type problem. However the fact that we are looking for the different combinations of 48 cheese sales instead of one has me stumped on what approach I should take.

So I need to figure out how many different types of orders have taken place, but I have no idea on how to start. Could anyone please help guide me in the right direction?

Since one of each cheese type has been sold, you know that an order must include one of each of the 34 cheese types. Now, since there are 14 (- 48 - 34) transactions unaccounted for, you could look at the different ways that the remaining 14 sales could come from the 34 cheeses.

 

[RoboNerd]

Hi, Thanks for replying.

I would disagree with you that each one of the cheese types must have been sold. It is possible, but it is also possible to have all of the 48 orders to be for one type of Feta Cheese, for example - as far as I see according to the wording of the problem.

If so, I do not understand where you are coming from in your approach..
Thanks!

 

[kuruman]

Look at the ends first. If all 34 kinds of cheese were sold, then the number of possible restocking orders is 1. If only one kind was sold, then the number of possible restocking orders is 34. Note that
$$1=\begin{pmatrix}
34\\
34
\end{pmatrix};~~~
34=\begin{pmatrix}
34\\
1
\end{pmatrix}
$$
What goes in between? However, I don't see the importance of 48 as opposed to any other number greater than 34 to guarantee the possibility that at least one of each kind was sold.

 

[RoboNerd]

Look at the ends first. If all 34 kinds of cheese were sold, then the number of possible restocking orders is 1. If only one kind was sold, then the number of possible restocking orders is 34. Note that
$$1=\begin{pmatrix}
34\\
34
\end{pmatrix};~~~
34=\begin{pmatrix}
34\\
1
\end{pmatrix}
$$
What goes in between? However, I don't see the importance of 48 as opposed to any other number greater than 34 to guarantee the possibility that at least one of each kind was sold.

Hi and thanks for your comment. Let's see what I can make here. So according to you we count the various possibilities for selling 34 types of cheese. That is a typical 'stars and bars problem' which I can do. However, the problem has 48 different orders.

Would that mean that I have 48 orders and I multiply it by the various possibilities for selling 34 types of cheese as in a typical stars and bars problem?

 

[kuruman]

However, the problem has 48 different orders.

The problem states 48 different sales not orders. I assume that at the end of the day the store owner looks at what is left and replenishes what is missing. It can be done with a single order if only one kind was sold, but there are 34 kinds so that the total number of possible restocking orders if only one kind has been sold is 34. That's how I read the question.

 

[Mark44]

I would disagree with you that each one of the cheese types must have been sold.

I misinterpreted this line:

Well I do know that each of the sales must involve buying at least one type of cheese

 

[RoboNerd]

The problem states 48 different sales not orders. I assume that at the end of the day the store owner looks at what is left and replenishes what is missing. It can be done with a single order if only one kind was sold, but there are 34 kinds so that the total number of possible restocking orders if only one kind has been sold is 34. That's how I read the question.

So - to paraphrase what you are saying to understand it correctly - you believe that a sale is equal to a single cheese item being sold?

 

[RoboNerd]

I misinterpreted this line:

No problem - fully OK and understandable!

 

[kuruman]

So - to paraphrase what you are saying to understand it correctly - you believe that a sale is equal to a single cheese item being sold?

Yes, a sale is a sale not to be confused with a sales receipt that may have more than one kind of cheese listed on it.

 

[RoboNerd]

Yes, a sale is a sale not to be confused with a sales receipt that may have more than one kind of cheese listed on it.

I am getting extremely confused. A sale could involve more than one types of cheese, but they ask me for how many restocking orders are possible.

If I sell Blue Cheese and Brie, then I have a different restocking order than if I sell Cheddar and Gouda, even though they all have the same number of cheese items being sold (2).

I am extremely confused. Any assistance would be greatly appreciated

 

[kuruman]

f I sell Blue Cheese and Brie, then I have a different restocking order than if I sell Cheddar and Gouda, even though they all have the same number of cheese items being sold (2).

Every time your cash register goes kachung, that's a sale. If you sell Blue Cheese and Brie that's two sales because your cash register goes kachung twice. If all you sell on a given day is just Blue Cheese and Brie you need one restocking order for that pair of cheeses because you sold nothing else. How many ways are there that you can sell only two kinds of cheese when you have 34 kinds? That's the number of possible restocking orders when you sell two different kinds. That number needs to be added to the number of restocking orders if you sold only one kind. Then you need to consider what happens when you sell three different cheeses and so on.

 

[RoboNerd]

So I consider find the all possible combinations for the type of cheese being sold. If I have more than one order being replicated, that is equal to the same restocking order.

So I try to find the combinations for the different types of cheese sold? OK. I follow that logic.

So it will be C(34,1) + C(34,2) + ... + C(34,34) = C(34 + 33, 33) by the bars and stars approach.

Is this approach right?

 

[kuruman]

Is this approach right?

That would be my approach.

 

[RoboNerd]

OK Thank you very much for the help!

 

[kuruman]

C(34,1) + C(34,2) + ... + C(34,34) = C(34 + 33, 33)

BTW, why is this equation valid?