Problem 30 - Discrete Mathematics with Applications

Question

Jane has two nickels, four dimes, three quarters, and two half-dollars in her handbag. Find the number of ways she can tip the waiter if she would like to give him: Not more than 50 cents, using only one type of coin.

Step-by-Step Solution

Answer

Jane can tip the waiter in 13 different ways, giving not more than 50 cents using only one type of coin.

Step 1: Calculate Amounts for Each Coin Type

Nickels: Since Jane has 2 nickels and each nickel is worth 5 cents, the maximum amount she can pay with nickels is 10 cents. Dimes: She has 4 dimes which are each worth 10 cents, so the maximum amount she can pay with dimes is 40 cents. Quarters: Jane has 3 quarters with each quarter being worth 25 cents. However, the ask is to not give more than 50 cents, so she can only use a maximum of 2 quarters, totaling 50 cents. Half-dollars: She has 2 half dollars, but each half-dollar is worth 50 cents, so she cannot give more than one half-dollar. Hence, she can give only one half-dollar or no half-dollars.

Step 2: Find the Number of Ways for Each Coin Type

Nickels: Jane can give 1 or 2 nickels or none at all. Thus, there are 3 ways for nickels. Dimes: Jane can give 1, 2, 3, or 4 dimes or none at all. So there are 5 ways for dimes. Quarters: As discussed earlier, Jane can give 1 or 2 quarters or none at all. So there are 3 ways for quarters. Half-dollars: She can give 1 half-dollar or none at all. So there are 2 ways for half-dollars.

Step 3: Calculate the Total Number of Ways

We will now add up the ways for each coin type: Total Number of Ways = Ways for Nickels + Ways for Dimes + Ways for Quarters + Ways for Half-dollars Total Number of Ways = 3 + 5 + 3 + 2 Total Number of Ways = 13 Jane can tip the waiter in 13 different ways, giving not more than 50 cents using only one type of coin.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many areas of mathematics and has many applications ranging from logic to statistical physics, from evolutionary biology to computer science, etc. In the context of our problem, combinatorics helps us find the number of ways Jane can tip the waiter with different combinations of coins, without exceeding the limit of 50 cents.

Combinatorial problems like the one Jane faces typically involve finding all the possible combinations that meet a given set of criteria, which is here, the different ways of combining nickels, dimes, quarter, and half-dollars to sum not more than 50 cents using only one type of coin. To simplify these problems, we employ various combinatorial principles such as the counting principle, permutations, and combinations.

Counting Principles

The counting principles, also known as the basic principles of counting, help to calculate the number of possible arrangements in a collection of items. Two of the most fundamental counting principles are the Addition Principle and the Multiplication Principle.

The Addition Principle states that if one event can occur in 'm' ways and a second independent event can occur in 'n' ways, then the total number of ways in which either event can occur is m + n. In our textbook problem, this is applied when we add up the number of ways Jane can tip using nickels, dimes, quarters, and half-dollars independently.

The Multiplication Principle states that if one event can occur in 'm' ways and a second event can occur in 'n' ways after the first event has occurred, then the total number of ways the two events can occur in sequence is m * n. This principle is not directly applied in the step-by-step solutions provided, as Jane is tipping with only one type of coin at a time, rather than a sequence of different coins.

Discrete Mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. This means that they are made up of distinct and separate elements, much like the coins in our problem. Coin combination problems are a perfect example of discrete mathematics because the coins are individual items that cannot be divided.

In solving discrete mathematical problems, we often use combinatorial analysis to count, arrange, and describe these discrete structures. In Jane's situation, the discrete elements are the coins, and we used combinatorial and counting principles to determine the number of ways she could use these discrete units to achieve a certain sum of money. Unlike continuous mathematics which deals with objects that can vary smoothly, discrete mathematics works with countable, distinct elements which is a key concept in understanding problems like the one posed in the textbook exercise.