# How do I solve this combinatoric/probability question regarding constrained allocation of asset.

I have $$N$$ bank account and $$B$$ USD savings. For $$i$$ in $$1, 2, 3, ..., N$$, the balance of bank account $$i$$, denoted as $$b_i$$, is subject to the constraint of $$l_i\leq b_i \leq u_i$$. The 1-year interest rate of account $$i$$ is $$r_i$$.

Notes:

• $$N$$, $$B$$, $$l_i$$, $$u_i$$ and $$r_i$$ are known constants.
• The minimum incremental unit of $$b_i$$ is 1 cent.
• The balance can be either credit or debit, i.e., $$b_i$$ can be either negative or positive as long as it satisfies the constraint.
• $$r_i$$ is the interest rate for both borrowing and lending.
• $$\sum_{i=1}^N b_i = B$$.
• $$\sum_{i=1}^N l_i <= B$$.
• $$\sum_{i=1}^N u_i >= B$$.

Questions:

1. How many possible ways are there to allocate my savings into the $$N$$ accounts? Or equivalently, how many different possible combinations of $$b_i$$ are there?
2. Suppose the distribution of different possible combinations of $$b_i$$ is a uniform distribution. Define a random variable $$R = \sum_{i=1}^N r_i\frac{b_i}{B}$$. So $$R$$ is the 1-year return of my savings. What does the distribution of $$R$$ look like? What is the mean and variance of $$R$$?
• What has been tried ?
– user645636
Mar 14, 2019 at 15:50
• I think my problem is a derivation of the "put balls in box" problem: math.stackexchange.com/questions/1441170/…. I am trying to solve a more complicated version of the "put balls in box" problem, which puts constraints on the number of balls a box can hold. I have not got any clues so far. Mar 15, 2019 at 15:36