If a set has 12 distinct elements, which expression represents the number of proper subsets of the set?

To find the number of proper subsets of a set with 12 distinct elements, you would start by calculating the total number of subsets, which is given by \(2^n\), where \(n\) is the number of elements in the set. Therefore, for a set with 12 elements, the total number of subsets is \(2^{12} = 4096\). However, proper subsets exclude the empty set and the set itself, so you would subtract 2 from this total. Thus, the expression for the number of proper subsets is \(2^{12} - 2\), which equals 4094. Now that we have the calculation, let’s keep the brain gears turning! Did you know that the concept of subsets dates back to set theory's formalization in the late 19th century? Mathematicians like Georg Cantor began exploring the nature of sets, leading to the foundation of modern mathematics! On to real-world applications: understanding subsets can make your life easier when organizing your collections or analyzing data! For instance, if you're planning a party and need to consider different combinations of guests (think of each guest as an element), subsets help you strategize seating arrangements or food preferences, ensuring you don’t unintentionally invite enemies!