Question

Iblauk lives in Baker Lake, Nunavut. She makes oven mitts to sell. She has wool duffel in red, dark blue, green, light blue, and yellow for the body of each mitt. She has material for the wrist edge in dark green, pink, royal blue, and red. How many different colour combinations of mitts can Iblauk make?

Short answer

20 different color combinations.

Step-by-step solution

Identify the Choices for the Body

Determine the number of color choices available for the body of each mitt. Iblauk has red, dark blue, green, light blue, and yellow wool duffel, which means she has 5 choices.

Identify the Choices for the Wrist Edge

Determine the number of color choices available for the wrist edge of each mitt. She has dark green, pink, royal blue, and red material for the wrist edge, which means she has 4 choices.

Calculate the Total Number of Combinations

Use the fundamental principle of counting to find the total number of different combinations. Multiply the number of choices for the body by the number of choices for the wrist edge: \[ 5 \text{ (colors for body)} \times 4 \text{ (colors for wrist edge)} = 20 \text{ (total combinations)} \]

Key concepts

Fundamental Principle of Counting

The fundamental principle of counting is all about figuring out how many possible outcomes you can get from a set of choices. If you have more than one step or stage, you simply multiply the number of choices at each step.

For example, imagine you're choosing an outfit. You have 3 shirts and 2 pairs of pants. You can think about each step separately: choose a shirt (3 options) and then choose a pair of pants (2 options).

To find out the total number of outfit combinations, you multiply the number of choices for each step: \[ 3 \text{ (choices for shirts)} \times 2 \text{ (choices for pants)} = 6 \text{ (total combinations)} \].

In Iblauk's mitt-making problem, we use the same principle. She has 5 colors for the mitt body and 4 colors for the wrist edge. Using the fundamental principle of counting, we multiply the choices: \[ 5 \text{ (body colors)} \times 4 \text{ (wrist edge colors)} = 20 \text{ (different combinations)} \].

Color Combinations

Color combinations involve mixing and matching different colors to create unique outcomes. This concept is particularly useful in design, fashion, and even in solving combinatorics problems.

Let's look at Iblauk's mitts. She has 5 color options for the mitt body: red, dark blue, green, light blue, and yellow. For the wrist edge, she has 4 options: dark green, pink, royal blue, and red.

To visualize, if she chooses red for the mitt body, she can then pair it with any of the 4 colors for the wrist edge. This process repeats for each color of the mitt body. Here's a simpler view of just one combination:

• If the body is red, the mitt can have: dark green, pink, royal blue, or red for the wrist edge.

This approach shows how different color combinations create various distinct looks.

Multiplication Rule in Probability

The multiplication rule in probability tells us how to find the likelihood of two independent events happening together. It is closely tied to the fundamental principle of counting.

To apply this rule, you multiply the probability of the first event by the probability of the second event. But right now, let's focus on its role in combinatorics.

In Iblauk's problem, each choice for the mitt body is independent of the choice for the wrist edge. That means the decision for one part doesn't affect the other.

We can use the multiplication rule to find total combinations. Since she has 5 choices for the body and 4 for the wrist edge, you multiply the number of choices: \[ 5 \text{ (body colors)} \times 4 \text{ (wrist edge colors)} = 20 \text{ (total combinations)} \].

This result shows the beauty of the multiplication rule in both counting and probability. It helps us understand how combining different choices multiplies the possibilities.