Distinguishable Permutations: A Simple Explainer Video

Welcome to understanding basic permutations! Today we'll explore how to arrange different objects in unique ways.Let's start by understanding what a factorial is, as it's crucial for calculating permutations.For example, when we have three different letters - A, B, and C - we can arrange them in various ways.Let's see all possible arrangements. First, we can start with ABC.Then we can swap B and C to get ACB.We can also start with B, giving us BAC and BCA.Finally, starting with C gives us CAB and CBA.Notice how we got exactly six different arrangements. This matches our factorial calculation of three factorial.There are three key points to remember about basic permutations.When choosing arrangements, we first select which element goes first. Then we arrange the remaining elements.This process of selecting elements one by one leads to the factorial formula, where we multiply the number of choices at each step.Now that we understand basic permutations, we're ready to explore more complex scenarios.When we have repeated elements in our sequence, we need to think differently about permutations.Let's look at the sequence AABB, where we have two A's and two B's.Here are some different ways we can arrange these letters.When we swap two identical letters, like the two A's, we get the same arrangement.Let's analyze how repeated elements affect our arrangements. When we swap identical letters, we don't create a new unique arrangement.This means that instead of having twenty-four different arrangements like we would with four different letters, we get fewer unique permutations.This is why we need a special formula to handle cases with repeated elements, which we'll explore next.For sequences with repeated elements, we need a special formula to calculate the number of distinguishable permutations.Let's use the word MISSISSIPPI as an example. This word has several repeated letters.First, we count the total number of letters, which is eleven.We then apply our formula, dividing eleven factorial by the factorial of each letter's frequency.Eleven factorial equals thirty-nine million, nine hundred sixteen thousand, eight hundred. We divide this by four factorial twice for I and S, two factorial for P, and one factorial for M.Multiplying the denominators gives us one thousand, one hundred and fifty-two.This gives us thirty-four thousand, six hundred and fifty unique arrangements.To understand how significant this reduction is, let's compare it to the number of arrangements if all letters were different.As you can see, accounting for repeated letters reduces the number of arrangements from nearly forty million to just thirty-four thousand, six hundred and fifty.