Problem 24 - Discrete Mathematics with Applications

Question

Find the number of ways each sum can be formed from a collection of 10 nickels and 5 quarters. 25 cents

Step-by-Step Solution

Answer

There are 2 ways to form 25 cents using the given collection of 10 nickels and 5 quarters: 1) using 5 nickels, and 2) using 1 quarter.

Step 1: Analyzing the possible combinations

To form a 25-cent sum, we can use only nickels, only quarters, or a combination of both. We will analyze each scenario.

Step 2: Using only nickels

Since a nickel is worth 5 cents, we can calculate how many nickels we need to form 25 cents. We have: \[\frac{25}{5} = 5\] So, one way to form 25 cents is to use 5 nickels.

Step 3: Using only quarters

Since a quarter is worth 25 cents, we simply need 1 quarter to form 25 cents. So, another way to form 25 cents is using 1 quarter.

Step 4: Using a combination of nickels and quarters

In this section, we need to find a combination of nickels and quarters that will give us a sum of 25 cents. Since we have already found that 1 quarter is equivalent to 25 cents, we do not need to consider any further combinations with quarters.

Step 5: Counting the number of ways

Now, let's count all the ways we have found to form 25 cents. 1. Use 5 nickels 2. Use 1 quarter We found a total of 2 ways to form 25 cents using the given collection of 10 nickels and 5 quarters.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arranging, and finding patterns in sets of elements. It is a foundational aspect of discrete mathematics and is essential for solving problems related to combinations and permutations. In the context of our problem, combinatorics helps us understand the number of ways we can form a particular sum (e.g., 25 cents) using different denominations of coins.

Consider the vast possibilities: if we have 10 nickels and 5 quarters, combinatorics allows us to systematically calculate the different ways we can select these coins to reach the desired sum. We use combinations when the order of selection does not matter (like in our coin problem), and permutations when the order is important - but since coins are indistinguishable in value when added, permutations are not required here.

Permutations and Combinations

Permutations and combinations are two key concepts in combinatorics that often confuse students. However, understanding their differences and applications is crucial for mastering problems like our coin example.

• Permutations refer to the different arrangements of a set of items where the order matters. Imagine arranging books on a shelf where the sequence is important.
• Combinations, on the other hand, involve selecting items from a set where the order does not matter—like picking teams from a group of players.

In our exercise dealing with nickels and quarters, we look for combinations of coins that total 25 cents. The order in which we place or select the coins doesn't affect the total sum, which is why we utilize the concept of combinations to solve it. The step-by-step solution correctly counts the possible selections (using all nickels or one quarter), reflecting the foundational principle of combinations—arranging items without regard to sequence.

Discrete Mathematics

Discrete mathematics is a vast field that includes topics such as graph theory, logic, combinatorics, and more. It focuses on discrete objects—think of them as countable items, like coins, students in a class, or computers on a network. Unlike continuous mathematics, which deals with objects that can vary smoothly (like calculus with infinitely divisible quantities), discrete mathematics works with items that are distinct and separate.

In the context of our coin example, discrete mathematics helps reason about finite sets of coins and find systematic ways to combine them to reach a sum. This kind of problem-solving is essential in computer science, economics, and other fields where making decisions based on distinct, countable assets is part of daily operations. The clarity and structure provided by discrete mathematics are key in creating efficient solutions for our textbook problem and beyond, guiding students to understand and manipulate finite or countable resources.