1. In an Olympic swimming event, two of the seven contestants are American. The contestants are randomly assigned to lanes 1 through 7. What is the probability that the Americans are assigned to the first two lanes?
2. Sixty-eight men's college basketball teams compete in the NCAA championship tournament. They are divided into four regions of seventeen teams each. The winners of each region advance to the Final Four. If a fan selects four teams at random, what are the odds against those teams being in the Final Four? Note: Of course, the fan will select only one team per region.
3. A collection of code words consists of all strings of seven characters, where each of the first three characters can be any letter or digit and each of the last four characters must be a digit. For example, 7A32765 is allowed, but 7A3B765 is not.
(a) What is the probability that a code word chosen at random begins with ABC?
(b) What is the probability that a code word chosen at random ends with 6578?
(c) What is the probability that a code word chosen at random consists of three letters followed by four even digits?