Make a list of all of the permutations of the letters A, B, C D, and E taken 3 at a time. How ma

Make a list of all of the permutations of the letters A, B,...

Key Concepts

Permutations

Permutations refer to the different arrangements of a set of objects where the order matters. In this context, determining the permutations involves arranging a subset of items from a larger set in a specific sequence.

Permutation Formula (nPk)

The permutation formula, expressed as P(n, k) = n! / (n - k)! where n is the total number of objects and k is the number of objects to arrange, provides a method to calculate the total number of ordered arrangements without listing them all. This formula is essential for quickly finding how many ways a subset can be arranged from a larger set.

Fundamental Counting Principle

The Fundamental Counting Principle is a key concept in combinatorics that states if there are multiple stages in an event, the total number of outcomes is the product of the number of choices at each stage. It is used to break down the process of creating sequences, such as arranging items, by multiplying the number of choices available at each step.

Factorial

Factorial, denoted by the exclamation mark (n!), is the product of all positive integers up to a given number n. It is used in permutation and combination formulas to account for the number of ways items can be arranged or ordered. Factorial values simplify the computation of expressions related to the total number of arrangements.

Transcript

00:02 All right, so here we have to list all the permutations of three out of five of these letters, and we also need to say how many permutations there should be in the list.

00:11 So i think it would be a good idea to start by figuring out the number of permutations in the list, and that way we'll know whether or not we miss some when we list them.

00:19 So we're only using three of the letters, and we have five choices for the first letter, four choices for the second letter, and three choices for the third letter.

00:27 So five times four times three is going to be 60.

00:30 So that means we're listing 60 different arrangements of letters.

00:34 Oh boy, here we go.

00:36 We're going to need to be really systematic about this and keep track of what we're doing.

00:41 So i'm going to start with a.

00:43 I could have a and then b and then c.

00:47 I could have a and then b and then d.

00:49 I could have a and then b and then e.

00:53 That takes care of my a and then my b.

00:56 What if i have a and then c and then b, a and then c, c and then d, a and then c and then e.

01:06 What if i have a and then d and then b, a and then d and then c, a and then d and then c, a and then d, a and then e, or what if i have a and then e and then e and then c, a and then e and then d? hopefully you see that i'm going in a systematic way, a with b, a with c, a with d.

01:30 There are 12 different arrangements listed so far, and 12 is one fifth of 60.

01:38 So we're going to have 12 arrangements with each starting letter that we work with.

01:42 So let's move on to the next 12.

01:48 Now i'm going to start with b.

01:49 It could be b and then a and then c, b and then a and then d, b and then a and then e.

01:57 Could be b and then c and then a, b, c, d, b, c, b, c, b, c, e.

02:04 Could be b -d -a, b -d -c, b -d -e.

02:12 Could be b -e -a, b -e -c -b -e -d...