Question

You have three groups of distinctly different items, four in the first group, seven in the second, and three 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

Identifying possible selections in each group

There are three groups of items. The first group has four items, the second group has seven items, and the third group has three items. We want to select one item from each group. Thus, we can perform 4 selections in the first group, 7 selections in the second group, and 3 selections in the third group.

Applying the counting principle

To find the total number of different triplets that can be formed, we can multiply the number of possible selections in each group. According to the counting principle: Number of different triplets = (Number of selections in the first group) * (Number of selections in the second group) * (Number of selections in the third group)

Calculating the result

Using the values from step 1, the number of different triplets that can be formed is: Number of different triplets = 4 * 7 * 3 Number of different triplets = 84 So, we can form a total of 84 different triplets by selecting one item from each group.

Key concepts

Combinatorics

Combinatorics is a branch of mathematics that studies counting, arrangement, and combination. In the context of our exercise, we are interested in finding out how many different ways we can choose items. Let's consider the definition of a combination. A combination is a selection of items where the order does not matter.

In the given problem, we focus on selecting triplets of items from different groups. The essential thing here is that each item from each group only gets combined with one item from the other groups. For example, if you pick the first item from the first group, you can combine it with any of the items from the second group and any from the third. This concept shows the importance of understanding how to count combinations in various contexts.

Multiplicative Principle

The Multiplicative Principle, also known as the Counting Principle, is a fundamental concept in combinatorics and is very handy for problems like the one we're solving. It states that if there are \(n\) ways to do one action and \(m\) ways to do another action, there are \(n \times m\) ways to perform both actions.

In this exercise, the principle is applied by first looking at each group's possible selections. If you select an item from the first group (4 ways), then for each choice, you have 7 possible selections in the second group. Finally, for each pair, you can choose from 3 items in the third group.

Thus, using the Multiplicative Principle, you calculate the total number of combinations by multiplying the number of selections from each group: \(4 \times 7 \times 3 = 84\). This shows how powerful and straightforward the principle can be when dealing with sequential choices.

Basic Probability

Basic probability often uses the principles from combinatorics to determine the likelihood of certain events. In this exercise, while we fundamentally apply counting concepts, these ideas cross over into probability.

When we talk about probability, we're considering the number of favorable outcomes over the total number of possible outcomes. While our problem doesn't directly ask for the probability, understanding how to calculate the total number of outcomes is an essential step when setting the stage for probability calculations.

Should this scenario involve some aspect of chance — for instance, wanting to know the probability of selecting a specific triplet — these calculated outcomes (84 triplets) provide the base for further probability calculations. Thus, learning combinatorial counting also builds a foundation for understanding and solving more complex probability problems.