# Finding the Number of Vectors bounded by r, n

1. Oct 3, 2011

### changeofplans

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

You are managing a virtual computer store and are responsible for maintaining the printer paper supply in r different locations. There is a total of n packets of printer paper in stock. To make sure printers do not run out of paper, you need to keep at least ci packets of paper in stock at location i. Assume n ≥ $\sum$ci.

a. How many different inventories (x1, x2, . . . , xr) are there?
b. Each location has also a maximal storage capacity di. Let r = 2 and, count the number of different inventories (x1, x2) with c1 ≤ x1 ≤ d1 and c2 ≤ x2 ≤ d2.

2. Relevant equations

We started doing sample spaces and events this week, so it looks like this is from the previous chapter, combinatorial analysis. I'm pretty sure this just uses combinations and a little logic.

3. The attempt at a solution

So I was looking at a few examples we did in class to compare this against; I understand that the inventories are just vectors that I have to count, and the number of vectors are bounded by r and n (locations and number of packets, respectively).

We can see each location as:

r1 + r2 + r3 + ... + ri

And then the packets of paper, c, need to each go into each of the locations such that:

r1 + r2 + r3 + ... + ri
^ ^ ^ ^
c1 c2 c3 ... cn

The issue is, i'm not really sure where to go from here, and i'm not even sure my approach is sound to begin with. Can anyone point me in the right direction? I'm a little lost :\

Thanks!