Problem 29 - 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: Exactly 50 cents.

Step-by-Step Solution

Answer

There are a total of \(\boxed{4}\) ways Jane can tip the waiter exactly 50 cents.

Step 1: List the available coins and their values

The coins Jane has in her handbag are: - 2 nickels (each worth 5 cents) - 4 dimes (each worth 10 cents) - 3 quarters (each worth 25 cents) - 2 half-dollars (each worth 50 cents)

Step 2: Calculate different ways to reach 50 cents

Let's analyze the possible combinations to make exactly 50 cents: 1. Using nickels (5 cents) and dimes (10 cents) only 2. Using only quarters (25 cents) 3. Using only half-dollars (50 cents)

Step 3: Find coin combinations for case 1

Let's examine case 1: using nickels and dimes to reach 50 cents. a. 5 nickels -> Jane only has 2 nickels, so this combination is not possible. b. 4 nickels and 1 dime -> Jane only has 2 nickels, so this combination is not possible. c. 3 nickels and 2 dimes -> Jane only has 2 nickels, so this combination is not possible. d. 2 nickels and 3 dimes -> this combination is possible since Jane has 2 nickels and 4 dimes. e. 1 nickel and 4 dimes -> this combination is possible since Jane has 2 nickels and 4 dimes. f. 5 dimes -> Jane only has 4 dimes, so this combination is not possible.

Step 4: Find coin combinations for case 2

Let's examine case 2: using only quarters to reach 50 cents. a. 2 quarters -> this combination is possible since Jane has 3 quarters.

Step 5: Find coin combinations for case 3

Let's examine case 3: using only half-dollars to reach 50 cents. a. 1 half-dollar -> this combination is possible since Jane has 2 half-dollars.

Step 6: Count possible combinations

We found the following combinations that reach exactly 50 cents: 1. 2 nickels and 3 dimes 2. 1 nickel and 4 dimes 3. 2 quarters 4. 1 half-dollar There are a total of \(\boxed{4}\) ways Jane can tip the waiter exactly 50 cents.

Key Concepts

Coin Combinations

When working with coin combinations, the central task is identifying which subsets of coins sum up to your target value. In Jane's problem, the goal is to use the coins she has to create exactly 50 cents. To effectively tackle such problems, follow these steps:

• List all coin types and their quantities.
• Identify all possible ways to combine these coins to reach the desired sum.
• Check each potential combination against the available number of coins to ensure its feasibility.

By systematically examining every possible option, you ensure no potential solution is overlooked. In this problem, Jane could combine her coins in four distinct ways that sum up to 50 cents, showcasing the diverse approaches to forming the same value from different coin combinations. This exercise reinforces the foundational principles of counting in combinatorics, giving rise to efficient problem-solving strategies.

Problem-Solving Strategy

Approaching combinatorics problems requires a robust and structured problem-solving strategy. The coin combination problem serves as a great example for applying such strategies effectively. Start by clearly understanding the problem and accurately listing given resources, such as the number of coins here. Then, categorize possible solutions by thinking through different cases or scenarios. For example, consider using:

• Only a single type of coin if possible (like only using quarters).
• Various combinations of the available types (like mixing nickels and dimes).

It's crucial to interpret constraints, for instance, the number of each coin Jane has, while evaluating each combination's feasibility. Finally, tally up valid combinations to determine the solution set. This methodical and segmented approach not only simplifies what initially might seem like a complex problem, but also ensures comprehensive coverage of potential solutions.

Discrete Mathematics

Discrete mathematics forms the backbone of combinatorial problems such as coin combinations. Unlike continuous math, where variables can take any value, discrete math deals with distinct, separate values. In Jane's problem, the distinct values correspond to the various coins and combinations. A core principle here is counting combinations while adhering to given constraints.

• Each combination considered needs to sum up to the exact target (50 cents in this case).
• The selection of each coin is limited to its available quantity.

Discrete mathematics helps guide systematic exploration of solution spaces, where you can't simply "fill in" gaps as you might in continuous systems. By understanding these mathematical foundations, we can better tackle real-world problems where resources are limited, discrete, and require critical inquiry to optimize or solve. Combinatorics, as a branch of discrete mathematics, shows us how to effectively and creatively analyze these limitations to find solutions.