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  1. The problem statement, all variables and given/known data

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?

  1. Relevant equations

Combination and permutation equations

  1. The attempt at a solution

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. Any help will be greatly appreciated

PhysNerd
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    Let $x_k$ be the number of sales of the $k$th type of cheese. Then the number of possible orders is the number of solutions of the equation $x_1 + x_2 + x_3 + \ldots + x_{34} = 48$ in the nonnegative integers. – N. F. Taussig Feb 26 '18 at 00:15
  • I am sorry but I do not understand why you say that there have been 48 cheese sales means that 48 packets of individual cheese (regardless of specific type) have been sold. Wouldn't the wording of the question mean that there were 48 total orders with varying amounts of different types of cheese? – PhysNerd Feb 26 '18 at 01:33
  • While your interpretation makes sense, I am not convinced there is enough information to solve the problem with your interpretation. – N. F. Taussig Feb 26 '18 at 03:07
  • Of course. Maybe the question is too ambiguous to be solved. – PhysNerd Feb 26 '18 at 03:10

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