a) Find a recurrence relation for the number of ternary strings of length n that do not contain two consecutive 0s or two consecutive 1s. b) What are the initial conditions? c) How many ternary strings of length six do not contain two consecutive 0s or two consecutive 1s?

Consider ternary strings-that is, strings where 0, 1, 2 are the only symbols used. For $n \geq 1,$ let an count the number of ternary strings of length n where there are no consecutive l’s and no consecutive 2’s. Find and solve a recurrence relation for an.

a) Find a recurrence relation for the number of bit strings of length n that contain a pair of consecutive 0s. b) What are the initial conditions? c) How many bit strings of length seven contain two consecutive 0s?

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 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?

In the Tower of Hanoi puzzle, suppose our goal is to transfer all n disks from peg 1 to peg 3, but we cannot move a disk directly between pegs 1 and 3. Each move of a disk must be a move involving peg 2. As usual, we cannot place a disk on top of a smaller disk. a) Find a recurrence relation for the number of moves required to solve the puzzle for n disks with this added restriction. b) Solve this recurrence relation to find a formula for the number of moves required to solve the puzzle for n disks. c) How many different arrangements are there of the n disks on three pegs so that no disk is on top of a smaller disk? d) Show that every allowable arrangement of the n disks occurs in the solution of this variation of the puzzle.

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

a) Find a recurrence relation for the number of ternary strings of length n that do not contain consecutive symbols that are the same. b) What are the initial conditions? c) How many ternary strings of length six do not contain consecutive symbols that are the same?

A palindrome is a string whose reversal is identical to the string. How many bit strings of length n are palindromes?

How many solutions does the equation

$$x_1 + x_2 + x_3 = 13$$

have where

$$x_1, x_2, and x_3$$

are nonnegative integers less than 6?

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.

What is Ksp expression?

Find the component form of v given its magnitude and the angle it makes with the positive x-axis. ǁvǁ = 8, θ = 60°