Problem 45 - Algebra 2

Question

customer in an ice cream shop can order a sundae with a choice of 10 flavors of ice cream, a choice of 4 flavors of sauce, and with or without a cherry on top. How many different sundaes are possible?

Step-by-Step Solution

Answer

There are 80 different possible sundaes.

Step 1: Identify choices for ice cream

The customer can choose 1 flavor out of 10 available flavors for the ice cream.

Step 2: Identify choices for sauce

The customer can choose 1 flavor out of 4 sauces available for the sundae.

Step 3: Consider cherry option

The customer has the option to include or exclude a cherry on top, providing 2 choices (with or without a cherry).

Step 4: Calculate total combinations

To find the total number of different sundaes, multiply the number of choices for ice cream, sauce, and cherry. Thus, the number of combinations is calculated as follows: \(10 \times 4 \times 2 = 80\).

Key Concepts

Multiplication Principle

The multiplication principle is a fundamental concept in combinatorics used to determine the total number of possible outcomes in a sequence of choices or events.
It states that if you have a number of ways to do one thing, and a number of ways to do another, the total number of ways to do both is the product of the two numbers.
For instance, in an ice cream sundae scenario, using the multiplication principle allows us to calculate the total options by examining each category of choice independently.

• 10 flavors of ice cream
• 4 flavors of sauce
• 2 choices for a cherry (with or without)

By multiplying these, the principle shows that all combinations can be found, resulting in a total of: \[ 10 \times 4 \times 2 = 80 \] This tells us there are 80 different sundae combinations available.

Probability

Probability is another important mathematical idea that measures how likely an event is to occur.
Although probability is not directly involved in calculating total combinations, understanding it helps in assessing the chances of making a particular choice. Imagine you pick a random sundae; knowing there are 80 possibilities provides context for potential individual selections.
If we were to calculate the probability of choosing a specific combination, it is the ratio of favorable outcomes to the total number of possibilities.
For a specific outcome, like selecting a chocolate ice cream with caramel sauce and a cherry, the probability would be: \[ P(\text{specific combination}) = \frac{1}{80} \]Understanding how many potential outcomes exist makes it clearer how probability can be applied to all sorts of practical decision-making scenarios.

Combinations

Combinations refer to the selection of items where the order does not matter. In our situation with the ice cream shop, this concept helps clear any confusion.
While we use the multiplication principle to calculate combinations of sundae options, remember, in this context, we are not concerned with the order in which the flavors, sauces, and cherry choices are made.
What matters is simply the choice itself.
For example:

• Choosing vanilla as the ice cream flavor
• Selecting hot fudge as the sauce
• Deciding on whether or not to have a cherry on top

Each choice is independent, and their overall sequence does not affect the final number. We only care about what is included in each distinct sundae flavor profile. This distinction is crucial when differentiating between permutations and combinations in broader mathematics.