Two-Bit Combinations

2B  Permutations

Take for example that you want to take a picture of your three friends Alice, Betty, and Cindy standing in a row for the camera. You, as the photographer, are free to arrange the women, starting from the left and moving clockwise around the frame. How many distinct orders are there for the women to be arranged in? Because there are only three women, it is not difficult to ponder for a moment and write down all of the possibilities; by referring to the ladies by the letters A, B, and C, we discover that there are six different possible arrangements.

Each of the possible configurations of the women is referred to as a "permutation," and there are a total of six permutations that are viable.

But let's say there were ten different ladies. Now the problem is more serious because it would take forever to write down all of the possibilities, and even if you tried, you would never know whether you left any out because you would never know which ones you missed. Therefore, it would appear that we are going to need to come up with a way to count the permutations without actually writing them all down. To accomplish this, first break the process of arranging the women in a line into manageable chunks, and then apply the concept of multiplication to each step.

To return to the example of the three ladies, let us suppose that there are three possible positions for the ladies to occupy:

There are three stages that correspond to one another in the process of arranging the ladies: first, we have to fill the first slot, then the second slot, and finally the third slot. There are three different ways in which a lady can occupy the first spot on the list. After this initial step is finished, there are two potential candidates for the second spot, given that there are still only two ladies remaining. After filling the first two slots, there is only one lady left, so there is only one way to fill the last slot. Finally, after filling the first two slots, there is only one lady left. According to the principle of multiplication, the number of different ways that the entire process can be finished by performing all three steps is

3 · 2 · 1 = 6

Obviously, the same method will work with ten ladies; the only difference will be that we will need ten slots and ten steps rather than three. There are a total of ten different combinations possible when ten women are used.

10 = 10 · 9 · 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 = 3,628,800

We refer to this using the abbreviation 10. referring to this extremely lengthy product In a more general sense, the notation N applies to any positive integer N. identifies the product in question.

N = N · (N−1) · (N−2) · … · 3 · 2 · 1   

We read "N , also known as "N factorial." Other factorials include the following:

1 = 1

2 = 2 · 1 = 2

3 = 3 · 2 · 1 = 6

4 = 4 · 3 · 2 · 1 = 24

5 = 5 · 4 · 3 · 2 · 1 = 120

6 = 6 · 5 · 4 · 3 · 2 ·1 = 720

A permutation of a group of items is a different way to arrange the items in a specific order. Because it is only logical that the ordering system we use for ladies ought to be applicable to other types of things as well, we have the following rule:

The First Rule of Permutations

The number of distinct permutations that are possible with N objects is equal to N.

The following examples will show that arranging things in a particular order is not the only way to do so; rather, there are a great deal of other options available.

example 1

A race has been entered by six different horses. How many different combinations of sequences are the horses able to finish the race in?

The ordering of the horses consists of a total of six steps. The winner of first place is determined first, followed by the runners-up for second, third, fourth, fifth, and sixth, respectively. There are six possible selections for first, followed by five for second, four for third, etc. In light of this, the total number of possible permutations is

6 = 6 · 5 · 4 · 3 · 2 ·1 = 720

example 2

There are five male students and five female students enrolled in a dance class. How many different waltz dance couples does the instructor have the ability to create?

The process of creating couples is broken down into five distinct stages by our company. To begin, we will number the ladies from one to five. There are five different ways that a woman can choose a male partner: number one For woman number 2, there are only four options left, for woman number 3, there are only three, for woman number 4, there are two, and for woman number 5, there is only one option. Lastly, for woman number 5, there is only one choice. The number of different ways a teacher can pair off 5 students is.

5 = 5 · 4 · 3 · 2 ·1 = 120

example 3

Nine players are selected to start for a baseball team. How many different ways are there for the coach to arrange the batting order?

The manager has nine different options for picking the leadoff batter, eight for picking the second batter, seven for picking the third batter, and so on. up until the very last batter, which can only be selected in one of two ways: The conclusion is that

9 = 9 · 8 · 7 · 6 · 5 ·4 · 3 · 2 · 1 = 362,880

example 4

The dog that belongs to Lori has given birth to four babes. Due to the fact that she cannot keep more than one dog at a time, she is going to give one puppy to each of her four cousins. How many different ways are there for her to split up the four puppies among her cousins?

The four steps are as follows: first, give Fido to a cousin in one of four different ways; next, give Lassie to another cousin in one of three different ways; finally, give Rover to a cousin in one of two different ways; and finally, give Spot to the final cousin. The total number of combinations that can occur is

4 = 4 · 3 · 2 · 1 = 24

example 5

A photograph will be taken of the quartet of acrobats. Every acrobat has the option of either standing on his or her feet or on their heads. How many different pictures are there to choose from?

Our permutation formula and the multiplication principle are both required to solve this problem. The procedure of lining up and positioning the acrobats can be broken down into five steps, which are as follows:

First step is to form a line with the four acrobats: = 24 ways

The second step is to position the first acrobat in one of two ways.

Step 3: Place the second acrobat in one of two positions

Step 4: Position the Third Acrobatic Performer in One of Two Ways

Step 5: Position the Fourth Acrobatic Performer in One of Two Ways

Therefore, the product is the total number of possible pictures.

24 · 2 · 2 · 2 · 2 = 384

If we wanted to go into more detail, we could divide step 1 into four smaller steps by lining up the acrobats one at a time. This would allow us to go into greater depth. On the other hand, given that we already have a formula for the procedure of lining up, we might as well use it.

Returning to the topic of acrobats, let us now assume that an acrobat troupe has a total of ten members. The publicist for the acrobat troupe will select four members of the troupe and arrange them for a publicity picture. For the sake of simplicity, they will all be standing in the same position. How many different pictures are there to choose from given these parameters? This problem is a little bit different from the permutation problems that came before it because this time we are not ordering all of the items in the group; rather, we are ordering a specific number of them. Despite this, the process of finding a solution to the issue is always the same: we break the process down into its component parts and apply the multiplication principle. We visualize four spaces in which the acrobats are going to stand, and we fill those spaces one at a time. There are ten different possibilities for filling the first slot, nine for the second, and so on. There are a total of ways to arrange four out of the ten acrobats that are available.

P(10,4) is the abbreviation for the product on the left, which can also be written out in full.

P(10,4) = 10 · 9 · 8 · 7 = 5040

P(10,4) is the notation used to denote the number of possible permutations when 10 items are shuffled four at a time.

In a more general sense, if the numbers N and M are both positive integers and N is greater than M, then P(N,M) represents the number of ways in which M objects can be chosen from N objects and placed in an order; the multiplication sign is used to calculate this number.

The formula for determining P(N,M) is as follows: P(N,M) = N (N1) (N2) ... (continue for M factors)

Here are examples:

P(7,3) = 7 · 6 · 5 = 210

P(12,4) = 12 · 11 · 10 · 9 = 11,880

P(4,1) = 4

P(9,5) = 9 · 8 · 7 · 6 · 5 = 15,120

P(100,2) = 100 · 99 = 9,900

P(6,6) = 6 · 5 · 4 · 3 · 2 · 1 = 720

The final illustration, P(6,6), demonstrates something that we are already familiar with, namely, that the number of possible ways to select N objects from N objects and arrange them in an order is.

P(N,N) = N

The formula for selecting one item out of N items is illustrated by the example P(4,1) = 4, which shows how the general formula works.

P(N,1) = N

In a nutshell, the second rule that applies to permutations is

The Second Rule of Permutations

P(N,M) represents the total number of possible permutations that can occur when M objects are selected from N objects.

example 6

There are five individuals competing for the position of quarterback on a football team. How many different ways does the head coach have to choose between the starting quarterback and the backup quarterback?

At this point, we are selecting two items from a total of five items, and then placing an order for those two items, with the first item serving as a starter, and the second item serving as a backup. There are 5 different ways to select the starter, and then there are 4 different ways to select the backup; therefore, the total number of permutations is 5.

P(5,2) = 5 · 4 = 20

example 7

How many different ways are there to award first, second, and third place ribbons in a race with seven female competitors?

There are a total of seven women competing, which means there are seven possible winners, six possible runners-up, and five possible third-place finishers. We are selecting three out of seven options and ranking them; there are total possible combinations.

P(7,3) = 7 · 6 · 5 = 210

example 8

In addition to her six professional outfits, Janet will spend the first three days of the following week attending a convention in Los Angeles related to her line of work. How many different combinations of business attire can she come up with for Monday, Tuesday, and Wednesday?

(a) if she alternates the suit that she wears every day

(b) if she is willing to wear the exact same suit on multiple occasions

In contrast to the second question, which is not a permutation question, the first one is. If she does wear a different suit every day, then we will need to select one suit for Monday, a different one for Tuesday, and yet another one for Wednesday. Because we have six different options for Monday, five for Tuesday, and four for Wednesday, the total number of permutations is.

P(6,3) = 6 · 5 · 4 = 120

In the second inquiry, we have the option of selecting the same answer each day; consequently, there are 6 possible answers for each day. According to the principle of multiplication, the number of possible combinations of the three options is

6 · 6 · 6 = 216

example 9

There are 5 women and 7 men in a dance class. To practice the tango, how many different ways can the teacher pair off the five students?

This issue can be described as being somewhat complex. The solution to this problem is to find a suitable partner for each woman. In step 1, a potential suitor is selected for Alice; in step 2, a potential suitor is selected for Betty; and so on. until each of the five women has a man in her life After the first step, there are 7 options for Alice, then 6 options for Betty, and so on; as a result, the total number of permutations after the first 5 steps is

P(7,5) = 7 · 6 · 5 · 4 · 3 = 2,520

If we try to do it the other way around, that is, select a woman for each man, then the method will not work because it is impossible to provide a woman for every single man. Furthermore, we are unable to simply solve the problem by assigning a woman to each of the five men because doing so would reduce the total number of men to five, which would change the nature of the issue. Furthermore, if we arbitrarily eliminate two of the men, we will eliminate some potential partners for couples.

example 10

Let's say that at the same time, four cars pull into a parking lot that otherwise has six spaces available. How many different ways are there for drivers to select parking spaces?

This tango problem is similar to the one that came before it. We have to make sure that cars are parked in the appropriate spots. Because there are more parking places than there are cars, we won't be able to put a car in each and every parking spot. However, we will make sure that each car has a parking spot, so we can assign the parking spots to the cars. There are four different options for the Chevy, five for the Ford, four for the Honda, and three for the Mazda. The Chevy has six options, while the Ford has five, the Honda has four, and the Mazda has three. The total number of possible combinations is.

P(6,4) = 6 · 5 · 4 · 3 = 360

EXERCISES 2B

1. How many different ways are there for six students to place their hamburger orders at the cafeteria?
2. How many different ways are there to assign four math professors to teach four different math classes if each professor is only responsible for one class?
3. There are currently nine male actors competing for roles in the upcoming production of The Three Musketeers. How many different actors does the director have to choose from in order to play Aramis, Athos, and Porthos?
4. How many different ways are there to arrange the three musketeers for a photograph?
   1. Each person has the option of donning or omitting their hat at all times.
   2. Each individual has the option of carrying or not carrying their respective swords, as well as the choice of whether or not to wear their hats.
   
   
5. Meg has a total of 12 cookbooks. How many different ways are there for her to arrange six of her books on the shelf that is located above her kitchen counter?

1. Amy's cat gave birth to a litter that consisted of three male and five female kittens. How many different options does Amy have?
   1. a female and a male kitten with the intention of gifting them to her niece.
   2. Give one of the kittens to each of her four cousins, which comes to a total of four.
   3. 2 male kittens, each of which she will give to one of her two sisters.
   4. a male kitten for Larry, a female kitten for Moe, and a male kitten for Curly
   5. Take a picture of the three male kittens and arrange them in a row.
   6. Three different kitten couples will perform the tango.

1. How many different ways are there for a group with 20 members to select their President, Vice President, Secretary, and Treasurer? (It is impossible for a single person to hold both positions. )
2. There are seven mailboxes at the apartment complex where the mailman is delivering four letters. If there are 7 mailboxes, how many different ways are there for the mailman to distribute the 4 letters?
   1. There is never more than one letter in any mailbox.
   2. Each mailbox is able to accommodate an unlimited amount of mail.
   
   
3. There are currently 11 parking spots available, but three cars have just pulled into the lot. Each vehicle has the option of either driving forwards or reversing into its designated parking spot. How many different routes are available to the three drivers?
   1. parking spaces
   2. spaces available for parking and parking positions

1. A changeup, a curveball, a slider, and a screwball are the other pitches in a pitcher's arsenal in baseball. How many different combinations of his next three pitches is he capable of choosing if
   1. He wants to change up his pitches as much as possible and not throw the same kind of pitch twice.
   2. He is willing to repeat what he has already said.
   
   
2. There are six starting pitchers for the University of Hawaii baseball team. How many different ways are there for the coach to choose his starting lineup for the team's three games against Arizona on Friday, Saturday, and Sunday?

This week, Sylvia can choose between four clean skirts and five clean blouses to wear to school. How many different ways are there for her to dress for work on Monday, Wednesday, and Friday, given that?

1. She does not want to repeat a skirt or blouse that she has already worn.
2. It is possible that she will wear the same thing again.
3. She is willing to recycle her dresses, but not her blouses or tops.