Suppose we have a door with a keypad lock; the lock has five buttons (labelled 1, 2, 3, 4, and 5) and a code for the lock is six digits long.
(a) What is the total number of possible combinations for the door? (b) What does the Pigeonhole Principle tell us about the possible combinations? (c) How many combinations are there that avoid having the same digit twice in a row? (d) Suppose we select a combination for the door at random. What is the probability that it consists only of odd numbers?