Compute total possible configurations in R

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I am looking for some help with finding an algorithm. I need to figure out how many possible configurations can be created using a list of resources for each configuration. Some configurations share resources, and some resources are unique to each configuration. For example, lets say I have 3 configurations (config1, config2, and config3 and resources A, B, C, D, E, F.

This table shows how many resources are in each configuration.

Resources Config1 Config2 Config3
A 0 1 3
B 1 4 2
C 3 2 0
D 1 1 2
E 0 0 3
F 6 6 6

And I have a table of the quantity on hand for each resource:

Resource On Hand
A 20
B 100
C 35
D 44
E 67
F 90

I believe this is an optimization problem, but I am not sure how to set it up for a large number of configurations (e.g., 20-40) and resources (e.g., 5000). Excel has an optimization solver, but cannot handle all of the resources. Is there such an algorithm for R?

Edit in answer to @on_an_island, if I throw this example into Excel Solver, the solution is 11, 0, 3 for Config1, 2 and 3. I also get an answer of 5 of each if I do not require solver to make unconstrained variables non-negative.

However, that doesn't give me the maximum of each configuration. If I simply divide the quantity on hand by each resource for each configuration, then the result is 11, 15, and 6 for Configs 1-3.

Resources Config1 Config2 Config3
A NA 20 6.667
B 100 25 50
C 11.667 17.5 NA
D 44 44 22
E NA NA 22.33
F 15 15 15


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