Exercise 6 (You Tryl) In the year 2007, Executive magazine surveyed the presidents of the 500 largest corporations in the United States. OF these 500 people, 310 had degrees (of any sort) in business, 238 had undergraduate degrees in business, and 184 had graduate degrees in business. How many presidents had both undergraduate and graduate degrees in business? Let \( U= \) undergraduate degree in Business. Let \( G= \) graduate degree in Business.

To solve the problem, we can use the principle of inclusion-exclusion. We are given the following information: - Total number of presidents surveyed, \( N = 500 \) - Number of presidents with undergraduate degrees in business, \( |U| = 238 \) - Number of presidents with graduate degrees in business, \( |G| = 184 \) - Number of presidents with any degree in business, \( |U \cup G| = 310 \) We need to find the number of presidents who had both undergraduate and graduate degrees in business, denoted as \( |U \cap G| \). According to the principle of inclusion-exclusion, we have: \[ |U \cup G| = |U| + |G| - |U \cap G| \] We can rearrange this equation to solve for \( |U \cap G| \): \[ |U \cap G| = |U| + |G| - |U \cup G| \] Now, we can substitute the known values into the equation: \[ |U \cap G| = 238 + 184 - 310 \] Let's calculate this step by step: 1. Calculate \( 238 + 184 \): \[ 238 + 184 = 422 \] 2. Now subtract \( 310 \) from \( 422 \): \[ 422 - 310 = 112 \] Thus, the number of presidents who had both undergraduate and graduate degrees in business is \( |U \cap G| = 112 \). Therefore, the final answer is: \[ \boxed{112} \]