To find the mood and figure of a categorical syllogism, we must first identify its three terms and then see how the middle term is positioned in the two premises.
The conclusion is 'No warships are commercial vessels.' The subject of the conclusion (the minor term) is 'warships', and the predicate of the conclusion (the major term) is 'commercial vessels'. The term common to both premises but absent from the conclusion (the middle term) is 'nuclear powered submarines'.
The premise containing the major term is 'No nuclear powered submarines are commercial vessels' - this is the major premise, of the form 'No M are P', an E-type proposition, with the middle term M as its subject.
The premise containing the minor term is 'All nuclear powered submarines are warships' - this is the minor premise, of the form 'All M are S', an A-type proposition, with the middle term M again as its subject.
Since the middle term is the subject of both the major premise and the minor premise (M-P, M-S), this arrangement is Figure III.
The conclusion 'No warships are commercial vessels' is itself an E-type proposition.
Reading the propositions in the standard order (major premise, minor premise, conclusion), we get E (major premise), A (minor premise), E (conclusion) - the mood EAE.
Combining the mood with the figure gives EAE, Figure III, which matches the required code.