Question

A marketing class did a survey of the number of fast-food outlets near campus. The results of the survey showed the following:$$\begin{array}{||l|c|}\hline \text { Type of Food Sold } & \text { No. of Outlets } \\\\\hline \text { Hamburgers } & 15 \\\\\hline \text { Tacos } & 25 \\\\\hline \text { Pizza } & 21 \\\\\hline \text { Hamburgers and tacos } & 11 \\\\\hline \text { Hamburgers and pizza } & 10 \\\\\hline \text { Tacos and pizza } & 14 \\\\\hline \text { Hamburgers and tacos and pizza } & 9 \\\\\hline \text { Served none of these items } & 5 \\\\\hline\end{array}$$ How many fast food outlets are there near campus?

Short answer

There are 40 fast food outlets near campus.

Step-by-step solution

Identify Sets and Their Intersection

Define sets for each type of food outlet: let \( H \) be the number of hamburger outlets, \( T \) be the number of taco outlets, and \( P \) be the number of pizza outlets. Also, record the intersections: \( H \cap T = 11 \), \( H \cap P = 10 \), \( T \cap P = 14 \), and \( H \cap T \cap P = 9\). Lastly, note that 5 outlets serve none of these items.

Apply Inclusion-Exclusion Principle

Use the formula for the union of three sets: \( |H \cup T \cup P| = |H| + |T| + |P| - |H \cap T| - |H \cap P| - |T \cap P| + |H \cap T \cap P| \). Here, substitute the values: \( 15 + 25 + 21 - 11 - 10 - 14 + 9 \).

Calculate Total Number of Outlets Serving One or More Items

Calculate the expression: \( 15 + 25 + 21 - 11 - 10 - 14 + 9 = 35 \). This value represents the number of outlets serving at least one type of food.

Total Fast Food Outlets Including Those Serving None

Add the number of outlets serving none of these items to the previous result: \( 35 + 5 = 40 \). This gives the total number of fast food outlets near campus.

Key concepts

Inclusion-Exclusion Principle

The Inclusion-Exclusion Principle is a fundamental tool in combinatorics for calculating the size of the union of multiple sets. It helps solve problems where elements are counted across overlapping categories. The principle can efficiently manage overlapping quantities by adjusting for double-counting or triple-counting of elements.

• To begin, the formula considers the sum of individual set sizes, which records all elements.
• Then, it subtracts the sizes of pairwise intersections, which take away the double-counted elements.
• Finally, it adds back the intersection of all three sets, as these were subtracted too many times.

In our problem, this means using the formula \( |H \cup T \cup P| = |H| + |T| + |P| - |H \cap T| - |H \cap P| - |T \cap P| + |H \cap T \cap P| \). The numerical application involves substituting set sizes and intersection counts, as shown in the solution.
Thus, applying it ensures the correct total outlets serving any of the specified foods. This principle is not just a formula to memorize but a logical process for resolving apparent overlapping contradictions in data.

Set Theory

Set theory is the mathematical study of collections of objects or numbers known as 'sets'. In this context, understanding set union, intersection, and complements are crucial for counting problems such as evaluating fast food outlets.

• A set can represent any collection of objects that share a characteristic, like hamburger outlets.
• The union of sets is a combination of all elements that appear in any of the included sets. In our case, all outlets serving any food type.
• The intersection of sets contains elements present in all included sets simultaneously, like outlets selling all three food types.

The survey data provided different intersections, such as those selling hamburgers and tacos, pizzas, or all three. Understanding these concepts lets us apply the Inclusion-Exclusion method correctly.
The simplicity of set theory's descriptive power makes it an indispensable mathematical tool for counting distinct categories accurately.

Problem Solving Strategies

Effective problem-solving in mathematics, especially involving set theory, relies on applying structured strategies. Here are some strategies demonstrated in the solution of this exercise:

• First, define your sets clearly. Identifying what each set represents clarifies the picture, as with \( H \), \( T \), and \( P \). This step sets up the groundwork.
• Next, record any available numbers and intersections from the problem statement. Listing out the intersections helps you see all overlapping areas.
• Then, employ a systematic approach like the Inclusion-Exclusion Principle. This provides an orderly method to tackle the overlap issue without missing any part of your solution.
• Finally, account for any additional elements outside the primary context, such as outlets serving none of the selected food types. Completion is key.

This strategy supports finding accurate solutions to complex counting problems by ensuring that no areas are overlooked. The real strength lies in using logical steps to deconstruct and then rebuild our understanding within a problem's structure.