SAT® question: Noun number agreement

[Ecoke0005]

Hello teachers.

(1)Puzzled, the children scratched their (head / heads).
(2)Many sons dislike their (fathers / father).
(3)The boys moved their (car / cars).

found this examples while surfing the Web.
The site explains like this.
(1) requires "head"; otherwise, the sentence implicates multi-headed childrenand.
(2) also requires "father" because students have only one father, and plural "fathers" indicates that sons have more than one fagther.
(3) could have either a car or cars because (a) plural implicates that each boy owned a car, and (b) singular implicates that boys (together) owned a car.

Is this explanation correct? As far as I know, singular nouns link to singular predicate, and plural nouns link to plural predicate. Therefore, all four cases need plural nouns in the predicate.

waiting for you replies...