Final answer:
Billy's mother's action of explaining why the broken toy was taken away is an example of the authoritative style of discipline, which involves setting limits and providing explanations.
Step-by-step explanation:
The mother's action in the given scenario where she explains her decision to take away the broken toy is an example of an authoritative style of discipline. In this style, the parent sets reasonable demands and consistent limits, while also expressing warmth and affection. They communicate with the child to explain the reasons behind rules and decisions, and are receptive to the child's point of view, as showcased by Billy's mother explaining why he can no longer play with the broken toy.