Math Home

Suppose you run \(k\) experiments. The first experiment has \(n_1\) outcomes. The second experiment has \(n_2\) outcomes no matter what the outcome was of the first experiment. The third experiment has \(n_3\) outcomes no matter what the outcomes of the first two experiments were, and so on until the \(k\)th experiment which has \(n_k\) outcomes no matter what the outcomes for the first \(k-1\) experiments were.

The multiplication rule says that the total number of possible outcomes for the \(k\) experiments is \(n_1 \cdot n_2 \cdot \dots \cdot n_k.\)

Example: When getting dressed in the morning, Lucy is deciding what to wear. She has \(3\) shirts, \(4\) pairs of pants, and \(6\) pairs of shoes. How many outfits could she make?

▼ Answer:

Each choice for an article of clothing can be considered as an experiment. The first experiment, the choice for the shirt, has \(3\) outcomes. The second experiment, the choice for the pants, has \(4\) outcomes. The third experiment, the choice for the shoes, has \(6\) outcomes.

By the multiplication rule, the total number of possible outcomes is \(3 \cdot 4 \cdot 6 = 72.\)

By the multiplication rule, the total number of possible outcomes is \(3 \cdot 4 \cdot 6 = 72.\)

At a bike shop, the size options include small, medium, and large. Any size can come in blue, green, red, or black. Every bike has the option of having a bell or not. How many bikes can you choose between if you only get to specify size, color and whether the bike has a bell?