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) Find a recurrence relation for the number of permutations of a set with n elements. b) Use this recurrence relation to find the number of permutations of a set with n elements using iteration.

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

Let Xₙ be the random variable that equals the number of tails minus the number of heads when n fair coins are flipped. a) What is the expected value of Xₙ? b) What is the variance of Xₙ?

A person deposits $1000 in an account that yields 9% interest compounded annually. a) Set up a recurrence relation for the amount in the account at the end of n years. b) Find an explicit formula for the amount in the account at the end of n years. c) How much money will the account contain after 100 years?

An employee joined a company in 1999 with a starting salary of $50,000. Every year this employee receives a raise of$1000 plus 5% of the salary of the previous year. a) Set up a recurrence relation for the salary of this employee $n$ years after 1999. b) What will the salary of this employee be in 2007? c) Find an explicit formula for the salary of this employee $n$ years after 1999.

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 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 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?

Comment essce qu'on va woyager dans le futur (in the funcre)? Fais un dessin de I'aéroport de l'avenir. Indique en fran$\c{c}$ais les personenes et les objets suivants dans ton dessin : a pilos, a stewardess, passengera, an aiplane, the boseling gaiv, the sehedule boatd, the currency exchange office.

Rembrandt Company acquired a plant asset at the beginning of Year 1. The asset has an estimated service life of 5 years. An employee has prepared depreciation schedules for this asset using three different methods to compare the results of using one method with the results of using other methods. You are to assume that the following schedules have been correctly prepared for this asset using (1) the straight-line method, (2) the sum-of-the-years'-digits method, and (3) the double-declining-balance method.

$$\begin{array}{cccccc} {\hspace{5pt}\text{}\hspace{5pt}}&&&\hspace{10pt}&\text{Sum-of-the-}&\text{\hspace{10pt}Double-declining}\\ \underline{\hspace{5pt}\text{Year}\hspace{5pt}}&&\underline{\hspace{5pt}\text{Straight-line}\hspace{5pt}}&\hspace{10pt}&\underline{\hspace{5pt}\text{Year's-Digits}\hspace{5pt}}&\hspace{10pt}\underline{\hspace{20pt}\text{Balance}\hspace{20pt}}\\ \text{1}&&\$9,000&\hspace{10pt}&\text{\$15,000}&\text{\$20,000}\\ \text{2}&&\hspace{5pt}9,000&&\text{\hspace{5pt}12,000}&\text{\hspace{5pt}12,000}\\ \text{3}&&\hspace{5pt}9,000&&\text{\hspace{10pt}9,000}&\text{\hspace{10pt}7,200}\\ \text{4}&&\hspace{5pt}9,000&&\text{\hspace{10pt}6,000}&\text{\hspace{10pt}4,320}\\ \text{5}&&\underline{\hspace{10pt}9,000\hspace{5pt}}&&\underline{\hspace{10pt}3,000}&\underline{\hspace{10pt}1,480}\\ \text{Total}&&\underline{\underline{\hspace{5pt}\$45,000}}&&\underline{\underline{\hspace{1pt}\$45,000}}&\underline{\underline{\hspace{1pt}\$45,000}}\\ \\ \end{array}$$

Instructions

Answer the following questions. a. What is the cost of the asset being depreciated? b. What amount, if any, was used in the depreciation calculations for the salvage value for this asset? c. Which method will produce the highest charge to income in Year 1? d. Which method will produce the highest charge to income in Year 4? e. Which method will produce the highest book value for the asset at the end of Year 3? f. If the asset is sold at the end of Year 3, which method would yield the highest gain (or lowest loss) on disposal of the asset?

Colonel Sporting Goods is authorized to issue 18,000 shares of common stock. During a two-month period, Colonel completed these stock-issuance transactions:

$$\begin{matrix} \text{Apr 23} & \text{Issued 3,000 shares of \$1.00 par common stock for cash of \$14.50 per share. }\\ \text{12} & \text{Received inventory valued at \$14,000 and equipment with market value of \$44,000 for 3,200 shares of the \$1.00 par common stock.}\\ \end{matrix}$$

Prepare the stockholders’ equity section of Colonel Sporting Goods’ balance sheet for the transactions given in this exercise. Retained Earnings has a balance of $48,000. Journal entries are not required.

Name three reasons why the colonists were at war.