Question

Use the mn Rule to find the number. There are two groups of distinctly different items, 10 in the first group and 8 in the second. If you select one item from each group, how many different pairs can you form?

Short answer

Answer: 80 different pairs can be formed.

Step-by-step solution

Calculate the number of ways to select from each group

We are given that there are 10 items in the first group and 8 items in the second group. Since we have to select one item from each group, there are 10 ways to choose an item from group 1 and 8 ways to choose an item from group 2.

Apply the mn Rule to find the total number of pairs

According to the mn Rule, we multiply the number of ways to choose an item from group 1 (10 ways) with the number of ways to choose an item from group 2 (8 ways) to find the total number of pairs. So, we have 10 * 8 = 80 different pairs that can be formed.

Key concepts

Combinatorics

Combinatorics is a field of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many practical applications in fields such as computer science, physics, and statistics.

In the context of the given exercise, combinatorics allows us to understand the different arrangements of items when we select one from each group. The crucial concept here is the combination of selections we can make when faced with multiple options. For instance, if you are at a restaurant and have to choose a starter and a main course from a menu, combinatorics helps you figure out how many unique meal combinations you can create.

It's not just about simple choices, either; combinatorics dives into more complex problems too, such as how many ways you can arrange people in a line, selecting committees from a group, or even solving jigsaw puzzles. It forms the foundational concept behind permutations and combinations, which are integral in solving many probability problems.

For students looking to deepen their understanding of such problems, it's essential to grasp the basics of factorial notation (which describes the number of ways to arrange a set number of items), permutations (arrangements of items where order matters), and combinations (selections of items where order doesn't matter).

Counting Principle

The counting principle, also known as the fundamental counting principle, is a guiding rule used to determine the number of possible outcomes in a given situation without having to list them all out. It's an essential concept in combinatorics and is quite useful in problems of probability.

Consider a simple analogy: you're picking out an outfit, and you have 3 shirts and 2 pairs of pants to choose from. The counting principle tells us we multiply the number of choices for shirts (3) by the number of choices for pants (2) to find that there are 6 different outfit combinations possible.

In the exercise provided, we applied this principle to find out how many different pairs of items could be formed from two distinct groups. By using the mn Rule (a direct application of the counting principle), we determined that selecting one item from each group (where 'm' is the number of choices for the first item and 'n' is the number of choices for the second item) the total number of unique pairs is the product of those two numbers of choices.

The counting principle is straightforward but powerful as it allows for the quick calculation of combinations without complicated or lengthy enumerations. It becomes especially useful when dealing with larger numbers or more complex scenarios.

Probability and Statistics

Probability and statistics are branches of mathematics that deal with data analysis and the prediction of outcomes. Probability is the measure of the likelihood that an event will occur, and statistics is the science of collecting, analyzing, interpreting, and representing numerical data.

In our everyday lives, we make decisions based on probable outcomes, like checking the weather forecast before deciding to carry an umbrella. The exercise in question involves fundamental concepts of probability where we determine the likelihood of various outcomes when we select items from different groups.

For example, if we take the solution to the given exercise, though not explicitly stated, there's an implicit understanding that each pair formed is equally likely to occur (having one item from the first group and one from the second). If we were instead asked what the probability of forming a particular pair is, we would divide the number of favorable outcomes (in this case, 1, since we're looking for one specific pair) by the total number of outcomes (80, as we calculated).

Understanding probability helps in interpreting results and making informed decisions; statistics use that understanding to collect concrete data and analyze it, which can be used in various fields like economics, psychology, and more. Together, they form a cornerstone of modern data analysis and interpretation.