Problem 3 - Combinatorics Through Guided Discovery

Question

One of the schools sending its team to the tournament has to send its players from some distance, and so it is making sandwiches for team members to eat along the way. There are three choices for the kind of bread and five choices for the kind of filling. How many different kinds of sandwiches are available? (b)

Step-by-Step Solution

Answer

15 different kinds of sandwiches are available.

Step 1: Identify the number of choices for bread

The problem states that there are three choices for the kind of bread.

Step 2: Identify the number of choices for filling

The problem states that there are five choices for the kind of filling.

Step 3: Calculate the total number of combinations

To find the total number of different kinds of sandwiches, multiply the number of choices for bread by the number of choices for filling: \[ 3 \times 5 = 15 \]

Key Concepts

Combinations

In mathematics, 'combinations' refer to the different ways in which a set of items can be selected, without regard to the order in which they are selected. In the context of sandwiches, each unique combination consists of one type of bread and one type of filling. Because the order doesn't matter (a sandwich with bread A and filling B is the same as a sandwich with filling B and bread A), we simply multiply the number of options. Here, we have 3 choices for bread and 5 choices for filling, producing a total of 15 unique combinations, calculated as follows:
\[3 \times 5 = 15\]
This combinatorial approach ensures we account for every potential unique sandwich combination possible with the given choices.

Multiplication Principle

The multiplication principle, also known as the Fundamental Principle of Counting, plays a vital role in solving combination problems. This principle states that if you have two independent choices to make, and the first choice can be done in 'm' ways, and the second can be done in 'n' ways, then there are \(m \times n\) ways in which both choices can be made. Here’s how it applies to our sandwich problem:
- **Bread choices:** 3 types
- **Filling choices:** 5 types
The multiplication principle tells us that to get the total number of sandwich options, we multiply the number of bread choices by the number of filling choices:
\[3 \text{ (breads)} \times 5 \text{ (fillings)} = 15 \text{ (sandwiches)}\]
This principle simplifies the problem-solving process by reducing complex combinations into straightforward multiplication.

Decision-Making Process

Breaking down problems into smaller steps can simplify the decision-making process and lead to a clearer understanding. In this exercise, the steps involve:
1. **Identifying the choices for bread**:
Understand and list that there are 3 types of bread.
2. **Identifying the choices for filling**:
Understand and list that there are 5 types of filling.
3. **Calculating the total number of combinations**:
Apply the multiplication principle by multiplying the number of bread options with the number of filling options.
This structured approach ensures all possible options are considered and accounted for, verifying that no combinations are overlooked, and leads to a methodical way of solving similar combinatorial problems in the future.