Question

Suppose you put one red marble, one green marble, and one blue marble in each of six bags. There are \(3^{6}\) possible orderings of the colors of the marbles you can get when you choose one marble from each bag. Suppose you put one red marble, one green marble, and one blue marble in each of six bags. There are \(3^{6}\) possible orderings of the colors of the marbles you can get when you choose one marble from each bag.

Short answer

There are \(3^{6}\) possible combinations of colored marbles that can be selected from each bag.

Step-by-step solution

Understand the set-up

Each of the six bags contains one marble each in three different colors - red, green, and blue. The task is to choose one marble from each bag.

Apply fundamentals of permutations

In permutations, the order matters. Here, we're looking for the number of ways the sequence of colors can be arranged when pulling one marble out of each bag. Since we're selecting one marble each from the six bags and each can be red, green, or blue, we use the multiplication principle for counting.

Use the power rule

The power rule states that the number of outcomes for an event raised to the number of occurrences will give the total number of possible outcomes. Here the event is choosing one of 3 differently colored marbles, and the occurrence is six times (once for each bag). Thus, the total number of possible outcomes is given by \(3^{6}\).

Key concepts

Multiplication Principle

When solving the problem of figuring out the total number of ways to pick marbles of different colors from multiple bags, we lean heavily on a basic concept in combinatorics known as the multiplication principle. This principle simply states that if an event can occur in 'm' ways and is followed by another event that can occur in 'n' ways, then the total number of ways the two events can occur in sequence is ‘m times n’—or simply, 'm * n'.

The multiplication principle can be extended to more than two events. In our marble example, we have six events (picking a marble from each of the six bags) and each event has 3 possible outcomes (picking either a red, green, or blue marble). Using the multiplication principle, we multiply the number of outcomes of each event together, thus: 3 outcomes/bag * 6 bags, rendering the multiplication of three, six times: which is represented mathematically as the expression ‘3 raised to the power of 6’, or in Latex, \(3^{6}\). Understanding this principle is crucial as it serves as the foundation for solving a wide variety of combinatorial problems.

Power Rule

The power rule is an integral part of understanding permutations particularly when the same event is repeated multiple times. This rule simplifies the process of multiplying a number by itself several times. The expression '3 raised to the power of 6', or \(3^{6}\), signifies that the number 3 is multiplied by itself 6 times.

This is central to our marble problem, where we are choosing one marble from each of six bags, with each choice being independent of the others and having three possible outcomes. Instead of writing out the number 3 multiplied by itself 6 times, we use the power rule notation for efficiency and clarity. The power rule is not just a shorthand; it also helps in recognizing patterns and calculating with exponents, which are frequent operations in algebra and combinatorics, particularly when dealing with permutations and combinations of various events.

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is related to many other areas of mathematics, such as algebra, probability, and geometry, and has applications in many fields like computer science, statistics, and logic.

To apply combinatorics to our marble example, we identify that the act of choosing marbles from the bags is a combinatorial problem where we look at the arrangements of the colors with respect to their positions—a fundamental topic in combinatorics. The problem requires us to explore the various permutations (possible orderings) of marble colors, which is a classic question in this field. Recognizing that there are three choices (red, green, blue) for the first position (the first bag), three for the second, and so on, we can use combinatorial reasoning to determine that each bag contributes to the possibilities exponentially, which is intuitively captured by the multiplication principle and translated into a precise calculation through the power rule.