This problem is based on the design of kidney exchanges discussed in class. There are five patients, each with a spouse who is a potential donor. The blood types of the patients and their prospective donors are: B-AB, A-B, 0-B, B-AB, and AB-0. Recall that an AB patient can accept any kidney, an A patient can accept an A or 0, a B patient can accept a B or 0, and a 0 patient can accept only a 0. Suppose there are no problems of tissue type incompatibility, and that all blood types are acceptable transplants are equally likely to be successful. What is the most number of transplants in principle that could be done in this group, and what are the patient-donor options for achieving this number? What is the most that can be done if donors only will donate if their spouse receives a transplant, and what options can achieve this maximum?