Problem 65 - Algebra and Trigonometry

Question

55–75 Solve the problem using the appropriate counting principle(s). Choosing a Pizza A pizza parlor offers four sizes of pizza (small, medium, large, and colossus), two types of crust (thick and thin), and 14 different toppings. How many different pizzas can be made with these choices?

Step-by-Step Solution

Answer

131,072 different pizzas can be made.

Step 1: Identify the Choices for Each Category

First, identify the number of choices available for each aspect of the pizza. The categories are size, crust type, and toppings. There are 4 different sizes (small, medium, large, colossus), 2 types of crust (thick, thin), and 14 different toppings.

Step 2: Calculate Possible Combinations for Size and Crust

To find the number of combinations of size and crust type, multiply the number of choices for size by the number of choices for crust. This gives us: \(4 \text{ sizes} \times 2 \text{ crusts} = 8 \text{ size and crust combinations}\).

Step 3: Calculate Possible Combinations for Toppings

Each pizza can have either no toppings or any combination of the 14 available toppings. For each topping, you have two choices: "include" or "not include." This results in \(2^{14}\) possible combinations for the toppings.

Step 4: Calculate Total Number of Different Pizzas

Finally, to find the total number of different pizzas, multiply the combinations obtained from size and crust by the combinations for the toppings. This gives us: \(8 \text{ size and crust combinations} \times 2^{14} \text{ topping combinations}\).

Step 5: Compute the Final Answer

Calculate \(2^{14}\), which is 16,384. Thus, the total number of different pizzas is \(8 \times 16,384 = 131,072\).

Key Concepts

Counting Principles

When solving problems related to how many ways something can be arranged or combined, counting principles are essential. A fundamental aspect of counting principles is the use of multiplication to determine the total number of outcomes when there are several independent choices. For example, in a problem where you need to choose among different options like pizza sizes and crust types, you can multiply the number of choices in each category to find all possible combinations.

In counting principles, it's important to identify each independent choice. Here, we had three categories: pizza size, crust type, and toppings. By understanding that each category's choice is independent, we can apply the multiplication principle to get a total number of combinations.

• 4 sizes
• 2 crust types
• Each choice is independent of others

This principle is powerful because it simplifies complex counting tasks into manageable steps.

Permutations and Combinations

In problems involving arrangements or selections, permutations and combinations play a significant role. With permutations, the order of selection matters, but when we talk about combinations, the order doesn't matter as much. However, in our pizza problem, we are more concerned with combinations.

The toppings represent a typical combination scenario. We have 14 different toppings and should determine how many ways we can combine them. For each topping, we decide whether to include it – that's a binary decision.

• Decide to include or not include each topping
• This results in a decision tree where each level doubles the possibilities (2 choices per topping)
• The total combinations of toppings can be found using the formula for combinations with two choices: \(2^{14} = 16,384\)

The number 16,384 includes pizzas with no toppings up to those with all 14. This simple rule of two allows us to quickly calculate it without complex formulas.

Mathematical Problem Solving

Mathematical problem solving combines logical thinking and numerical calculations. It is essential for finding solutions accurately. Let's dissect the pizza problem using problem-solving strategies:

First, break down the problem into smaller parts or categories, like size, crust, and toppings. This method makes the problem less overwhelming. Assess each category independently to see how they contribute to the whole.

• Size and Crust: Direct multiplication to find all combinations
• Toppings: Apply binary combinations
• Combine results from all smaller problems

Finally, combine all the solutions for a complete answer, multiplying the outcomes of each category to get the total. In structured problem solving, it's critical to double-check math calculations like powers and multiplications. This ensures no step is overlooked, finalizing an answer of 131,072 different pizzas. Through logical sequencing, and application of mathematical concepts, you resolve even complex problems efficiently.