a) Find a recurrence relation for the number of bit strings of length n that do not contain three consecutive 0s. b) What are the initial conditions? c) How many bit strings of length seven do not contain three consecutive 0s?

A country uses as currency coins with values of 1 peso, 2 pesos, 5 pesos, and 10 pesos and bills with values of 5 pesos, 10 pesos, 20 pesos, 50 pesos, and 100 pesos. Find a recurrence relation for the number of ways to pay a bill of n pesos if the order in which the coins and bills are paid matters.

a) Explain how to find a recurrence relation for the number of bit strings of length n not containing two consecutive 1s. b) Describe another counting problem that has a solution satisfying the same recurrence relation.

a) Find a recurrence relation for the number of ways to climb n stairs if the person climbing the stairs can take one stair or two stairs at a time.

b) What are the initial conditions?

c) In how many ways can this person climb a flight of eight stairs?

a) Find a recurrence relation for the number of ternary strings of length n that contain two consecutive 0s.

b) What are the initial conditions?

c) How many ternary strings of length six contain two consecutive 0s?

A bus driver pays all tolls, using only nickels and dimes, by throwing one coin at a time into the mechanical toll collector. a) Find a recurrence relation for the number of different ways the bus driver can pay a toll of n cents (where the order in which the coins are used matters). b) In how many different ways can the driver pay a toll of 45 cents?