Find the number of proper subsets of the set. \( \{x \mid x \) is an even number between 17 and 29\( \} \)

To determine the number of **proper subsets** of the given set, let's follow these steps: 1. **Identify the Elements of the Set:** The set is defined as: \[ \{x \mid x \text{ is an even number between } 17 \text{ and } 29\} \] - **Even numbers between 17 and 29** (exclusive) are: 18, 20, 22, 24, 26, and 28. - So, the set can be written as: \(\{18, 20, 22, 24, 26, 28\}\). 2. **Determine the Number of Elements:** - There are **6 elements** in the set. 3. **Calculate the Total Number of Subsets:** - For a set with \( n \) elements, the total number of subsets is \( 2^n \). - Here, \( 2^6 = 64 \) subsets. 4. **Determine Proper Subsets:** - A **proper subset** is any subset that is **not equal** to the original set. - Therefore, the number of proper subsets is the total number of subsets minus one (to exclude the set itself). - So, \( 64 - 1 = 63 \). **Final Answer:** There are **63** proper subsets of the set.