 If we take a look at the three letters A, B, and C, there are eight different possible ways or permutations to combine these letters. There are 16 different ways to arrange A, B, C, and D. In fact, the number of ways to arrange these letters will keep doubling forever. A, B, C, D, and E have 32 different possible arrangements. All of these subsets of these sets of letters are called power sets. Note that order doesn't matter, so I'm always listing these combinations in alphabetical order. Let's see if you can prove why this doubling will continue forever. Focusing on the case of four letters, notice that all of the combinations with three letters are also valid combinations of four letters, since we don't need to use the final letter. But for every existing combination without the last letter, we can create a new combination WITH the last letter. So every time we add a letter, we can create a new set of combinations, each one with a new letter added. Hence, the number of combinations will double each time, and given that we have a single letter resulting in two combinations, we can use induction to infer that the number of combinations of N letters is 2 to the power N. Thanks for watching.