A hospital operating room needs to schedule three knee surgeries and two hip surgeries in a day. Suppose that an operating room needs to schedule three knee, four hip, and five shoulder surgeries. Assume that all schedules are equally likely. Determine the following probabilities:

(a) All hip surgeries are completed first given that all knee surgeries are last.

(b) The schedule begins with a hip surgery given that all knee surgeries are last.

(c) The first and last surgeries are hip surgeries given that knee surgeries are scheduled in time periods 2 through 4.

(d) The first two surgeries are hip surgeries given that all knee surgeries are last.