Short Proof: Exact Solution to the Finite Frobenius Coin Problem

The Frobenius Coin Problem is a classic question in mathematics: given coins of specified denominations, what is the largest amount that cannot be formed using only those coins? This brief work covers a variation of such question, posing a limit on the number of coins available for each denomination. Thus, the new problem becomes finding the count of distinct values that can be represented, and those that cannot, within the finite set of integers ranging from zero to the sum of all coins. We refer to this version of the problem as the "finite" case. We will show how this closely relates to the original question, and prove an exact formula solving the problem when exactly two denominations are involved.