A hedge fund manager needs to construct a portfolio by selecting 30 stocks from 20 different sectors. Each sector contains exactly 4 different stocks. The manager aims to maximize the number of sectors from which at least 3 stocks are included in the portfolio. Additionally, even though the manager is not aware of the benefits of diversification and wants to minimize his risk exposure across multiple different stock sectors, sometimes there are market constraints that force the selection of some sectors in the portfolio with 4, 3, 2, or 1 stocks. For example, if 5 sectors are selected with all 4 stocks each, then 10 more stocks need to be chosen from the remaining sectors, possibly selecting 3 stocks from 2 sectors, and 2 stocks from two other sectors.

How many different portfolios can the manager create under these conditions?

Hint: Start by considering the maximum and minimum possible numbers of sectors with 3 or more stocks. Use this to determine the possible range for the number of full sectors (sectors with 4 stocks) and express the possible number of full sectors parametrically (in terms of a variable). Then, count all the ways to select portfolios by choosing parametrically the number of full sectors and filling