Question

Use the mn Rule to find the number. There are three groups of distinctly different items, 4 in the first group, 7 in the second, and 3 in the third. If you select one item from each group, how many different triplets can you form?

Short answer

Answer: 84 different triplets can be formed.

Step-by-step solution

First Group Selection

In the first group, there are 4 items. Since we need to select one item from this group, we have 4 different possibilities.

Second Group Selection

In the second group, there are 7 items. Similarly, since we need to select one item from this group, we have 7 different possibilities.

Third Group Selection

In the third group, there are 3 items. As with the previous groups, we need to select one item from this group, which gives us 3 different possibilities.

Applying the mn Rule

Now, according to the mn Rule, we multiply the number of possibilities for each group to find the total number of different triplets: 4 (possibilities from first group) * 7 (possibilities from second group) * 3 (possibilities from third group) = 84. So, there are 84 different triplets that can be formed by selecting one item from each group.

Key concepts

Combinatorics

Combinatorics is an area of mathematics that focuses on counting, arranging, and combining objects. It’s an essential part of understanding how many possible outcomes or configurations can occur given a certain set of rules. In our textbook exercise, we delve into a combinatorial principle known as the mn Rule, which simply states that if we have m ways to do something and n ways to do another thing, then there are m*n ways to perform both actions.

For example, imagine choosing a pair of shoes and a hat to wear. If you have 5 pairs of shoes and 4 hats, you have 5*4 = 20 different combinations. Similarly, in our exercise, the student must use the mn Rule to find out how many different triplets can be formed from three groups of items. As clarified by the steps provided, by multiplying the number of choices in each group (4, 7, and 3), the student arrives at the solution of 84 different triplets. Understanding the mn Rule is fundamental in combinatorics because it lays the groundwork for solving more complex problems and is often the first step in determining the probability of certain outcomes.

Probability and Statistics

The field of probability and statistics is concerned with the analysis and interpretation of data and the chances of various outcomes. It’s widely used in fields such as science, economics, and engineering. When solving problems related to probability, we often deal with the likelihood of certain events occurring, given a finite set of possibilities.

In combinatorics, such as the mn Rule from our exercise, we don't always immediately discuss probabilities. However, combinatorial principles are at the heart of calculating probabilities. Once you know, for example, that there are 84 different triplets that can be formed, you can begin to ask questions like, 'What is the probability that a randomly selected triplet will contain a specific item from one of the groups?' Understanding the total number of outcomes is key because the probability of a single event is usually calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

Permutations

Permutations are arrangements of objects in a specific order, and finding them is an important aspect of combinatorics. The concept of permutations is often illustrated by asking how many different ways you can arrange a set of distinct items. For instance, how many ways can you arrange 3 books on a shelf? The answer would be a permutation calculation.

The textbook exercise we are referring to is related to permutations in that we are selecting one item from each group and arranging them into a triplet, which is a sequence with a particular order. However, because the order in which we select items from each group does not matter in this exercise, we're not calculating permutations in the strictest sense. Permutations become significant when the order of selection does matter, like determining first, second, and third place in a race among a group of runners. Understanding when and how to use permutation calculations is crucial for many combinatorial problems, particularly in games, scheduling, and other scenarios where the sequence of events or selections is paramount.