Topic: Discrete Mathematics "Advanced Counting Techniques"

1. The shellfish population in a bay is estimated to be a million in the year 2000. Studies show that pollution reduces this population by about 2% per year, while other hazards are judged to reduce the population by 10,000 per year. Show how you derive a recurrence system for the shellfish population in the n-th year after 2000.

2. A gambler repeatedly bets $1 that a fair coin will come up heads when tossed. Each time the coin comes up heads, the gambler wins $1; and each time it comes up tails, he loses $1. The gambler will quit playing either when he is ruined (loses all his money) or when he has $M (where M is a value he has decided in advance). What is the probability that the gambler is ruined when he begins playing with $n. Formulate a recurrence system for it.

3. Let = {0, 1, 2} be an alphabet. Show how you derive a recurrence system for the number of ternary strings x * that contain at least one occurrence of consecutive 0's or consecutive 1's