This paper investigates finite-time (FT) stability and stabilization problems for a class of switched linear systems with polytopic uncertainties. Both stable and unstable subsystems are considered to coexist in the system, and a new concept of extended FT stability is proposed as the first attempt. A stability criterion is first established, where the admissible maximum switching number is obtained while ensuring extended FT stability of switched linear systems with time-varying delays under a given maximum ratio between the running time of unstable subsystems and the running time of stable subsystems. Sufficient conditions on the existence of desired memory state-feedback controllers are then developed. A numerical example and a class of servomechanism systems are given, respectively, to illustrate the effectiveness and validity of the developed techniques with time-varying delays and without time delay.