The given plasmid is a circular DNA molecule, and it contains two recognition sites: one for HindIII and one for EcoRI. Both enzymes will cut the plasmid at their respective recognition sites. Since both HindIII and EcoRI cut the plasmid at distinct positions, we will have one cut at each site. The total number of fragments produced will depend on how many times each enzyme cuts.
The plasmid is cut at two positions, one by HindIII and one by EcoRI. This results in two cuts in total, dividing the plasmid into four fragments.
Thus, the total number of DNA fragments produced after complete digestion by both enzymes is 4.