The Multiplication Principle can be used to solve a variety of problem types. One type of problem involves placing objects in order. We arrange letters into words and digits into numbers, line up for photographs, decorate rooms, and more. An ordering of objects is called a permutation.
Finding the Number of Permutations of n Distinct Objects Using the Multiplication Principle
To solve permutation problems, it is often helpful to draw line segments for each option. That enables us to determine the number of each option so we can multiply. For instance, suppose we have four paintings, and we want to find the number of ways we can hang three of the paintings in order on the wall. We can draw three lines to represent the three places on the wall.
The permutation (1 2)(3 4) in A 4 shows that the converse is not true in general. Equivalence of the two definitions. This section presents proofs that the parity of a permutation σ can be defined in both ways: As the parity of the number of inversions in σ (under any ordering),. First number is 123456 for sure ryt? Lets find out how many numbers are formed starting with 1:. Starting with 1 =5! = 120 So 124th number will surely start with 2. Starting with 2134 = 2 (i.e 265). Starting with 2135 =. Create a Permutation object with 12 elements and perform Permute randomly (blocks): 0, 0, 4, 'yes', 'yes'. We randomly permute blocks of size 4 and permute randomly within these blocks and make sure that on the transition from on block to the other no two stimuli are equal. Permute 3.4.2 MacOS Full Permute Video, audio and image files come in many different kinds and shapes, but sometimes you need a specific format since your iPad or DVD player won’t play that video. For example, the array 3, 2, 1, 0 represents the permutation that maps the element at index 0 to index 3, the element at index 1 to index 2, the element at index 2 to index 1 and the element at index 3 to index 0.
There are four options for the first place, so we write a 4 on the first line.
After the first place has been filled, there are three options for the second place so we write a 3 on the second line.
After the second place has been filled, there are two options for the third place so we write a 2 on the third line. Finally, we find the product.
There are 24 possible permutations of the paintings.
How To: Given [latex]n[/latex] distinct options, determine how many permutations there are.
- Determine how many options there are for the first situation.
- Determine how many options are left for the second situation.
- Continue until all of the spots are filled.
- Multiply the numbers together.
Example 2: Finding the Number of Permutations Using the Multiplication Principle
Kcncrew pack 2015 07 15 download free. At a swimming competition, nine swimmers compete in a race.
- How many ways can they place first, second, and third?
- How many ways can they place first, second, and third if a swimmer named Ariel wins first place? (Assume there is only one contestant named Ariel.)
- How many ways can all nine swimmers line up for a photo?
Solution
- Draw lines for each place.There are 9 options for first place. Once someone has won first place, there are 8 remaining options for second place. Once first and second place have been won, there are 7 remaining options for third place.Multiply to find that there are 504 ways for the swimmers to place.
- Draw lines for describing each place.We know Ariel must win first place, so there is only 1 option for first place. There are 8 remaining options for second place, and then 7 remaining options for third place.Multiply to find that there are 56 ways for the swimmers to place if Ariel wins first.
- Draw lines for describing each place in the photo.There are 9 choices for the first spot, then 8 for the second, 7 for the third, 6 for the fourth, and so on until only 1 person remains for the last spot.There are 362,880 possible permutations for the swimmers to line up.
Analysis of the Solution
Note that in part c, we found there were 9! ways for 9 people to line up. The number of permutations of [latex]n[/latex] distinct objects can always be found by [latex]n![/latex].
![Permute 3 4 4 equals equal Permute 3 4 4 equals equal](https://treepx.com/wp-content/uploads/2020/06/Permute-3.4.11.www_.treepx.jpg)
A family of five is having portraits taken. Use the Multiplication Principle to find the following.
Try It 3
How many ways can the family line up for the portrait?
Try It 4
How many ways can the photographer line up 3 family members?
Try It 5
How many ways can the family line up for the portrait if the parents are required to stand on each end?
Finding the Number of Permutations of n Distinct Objects Using a Formula
For some permutation problems, it is inconvenient to use the Multiplication Principle because there are so many numbers to multiply. Fortunately, we can solve these problems using a formula. Before we learn the formula, let’s look at two common notations for permutations. If we have a set of [latex]n[/latex] objects and we want to choose [latex]r[/latex] objects from the set in order, we write [latex]Pleft(n,rright)[/latex]. Another way to write this is [latex]{n}_{}{P}_{r}[/latex], a notation commonly seen on computers and calculators. To calculate [latex]Pleft(n,rright)[/latex], we begin by finding [latex]n![/latex], the number of ways to line up all [latex]n[/latex] objects. We then divide by [latex]left(n-rright)![/latex] to cancel out the [latex]left(n-rright)[/latex] items that we do not wish to line up.
Let’s see how this works with a simple example. Imagine a club of six people. They need to elect a president, a vice president, and a treasurer. Six people can be elected president, any one of the five remaining people can be elected vice president, and any of the remaining four people could be elected treasurer. The number of ways this may be done is [latex]6times 5times 4=120[/latex]. Using factorials, we get the same result.
[latex]frac{6!}{3!}=frac{6cdot 5cdot 4cdot 3!}{3!}=6cdot 5cdot 4=120[/latex]
There are 120 ways to select 3 officers in order from a club with 6 members. We refer to this as a permutation of 6 taken 3 at a time. The general formula is as follows.
[latex]Pleft(n,rright)=frac{n!}{left(n-rright)!}[/latex]
Note that the formula stills works if we are choosing all [latex]n[/latex] objects and placing them in order. In that case we would be dividing by [latex]left(n-nright)![/latex] or [latex]0![/latex], which we said earlier is equal to 1. So the number of permutations of [latex]n[/latex] objects taken [latex]n[/latex] at a time is [latex]frac{n!}{1}[/latex] or just [latex]n!text{.}[/latex]
A General Note: Formula for Permutations of n Distinct Objects
Given [latex]n[/latex] distinct objects, the number of ways to select [latex]r[/latex] objects from the set in order is
[latex]Pleft(n,rright)=frac{n!}{left(n-rright)!}[/latex]
How To: Given a word problem, evaluate the possible permutations.
- Identify [latex]n[/latex] from the given information.
- Identify [latex]r[/latex] from the given information.
- Replace [latex]n[/latex] and [latex]r[/latex] in the formula with the given values.
- Evaluate.
Example 3: Finding the Number of Permutations Using the Formula
A professor is creating an exam of 9 questions from a test bank of 12 questions. How many ways can she select and arrange the questions?
Solution
Icarefone 2 2 1 download free. Substitute [latex]n=12[/latex] and [latex]r=9[/latex] into the permutation formula and simplify.
[latex]begin{array}{l}text{ }Pleft(n,rright)=frac{n!}{left(n-rright)!}hfill Pleft(12,9right)=frac{12!}{left(12 - 9right)!}=frac{12!}{3!}=79text{,}833text{,}600hfill end{array}[/latex]
There are 79,833,600 possible permutations of exam questions!
Analysis of the Solution
5 Permute 3
![Permute Permute](https://prod-qna-question-images.s3.amazonaws.com/qna-images/question/874141f2-494d-4a75-aa24-1ee87a425b6e/9887010c-af75-4b5f-9a61-c1de682dcf86/il2c51a_processed.jpeg)
We can also use a calculator to find permutations. For this problem, we would enter 15, press the [latex]{}_{n}{P}_{r}[/latex] function, enter 12, and then press the equal sign. The [latex]{}_{n}{P}_{r}[/latex] function may be located under the MATH menu with probability commands.
Q & A
Could we have solved using the Multiplication Principle?
Yes. We could have multiplied [latex]15cdot 14cdot 13cdot 12cdot 11cdot 10cdot 9cdot 8cdot 7cdot 6cdot 5cdot 4[/latex] to find the same answer.
A play has a cast of 7 actors preparing to make their curtain call. Use the permutation formula to find the following.
Try It 6
How many ways can the 7 actors line up?
Try It 7
How many ways can 5 of the 7 actors be chosen to line up?
Permute! version 3.4 alpha 9
User's manual
Multiple regression over distance, ultrametric and additive matrices with permutation test
Last revision: Saturday, March 30, 2013
Version History
Download it hereImportant
Permute! 3.4 is a major update to the Permute! 3.2 program, despite the small change in version number. It has been made easier to use, faster and more error-tolerant. Hopefully you will find this program to be highly usable, much more than the previous version. This is alpha software, which means it is usable and the feature list is almost complete but not totally implemented.
What does Permute! do?
This program computes a multiple regression of one or more independent variables over one dependent variable, and assesses the probability (p-value) of the regression coefficients and the associated R2 (R-squared) using a permutational method described in Legendre, Lapointe and Casgrain (1994). It also allows the user to select the variables that contribute most to the variation through several procedures: backward elimination, forward selection and stepwise regression.The program is remarkable in its use of appropriate permutation methods. If the variables represent distance matrices, ultrametric or additive trees, the permutation method accounts for it by generating a permuted distance matrix or tree.
How to use the program
Keeping with the philosophy of Macintosh programs, the author assumed that users do not like to read manuals. Hence the program's interface is as self-explanatory as possible, and has balloon help throughout. However, it assumes that the user is familiar with the contents of the Legendre, Lapointe and Casgrain (1994) paper, or at least with the concepts of multiple regression. To start the program, double-click on the icon. From the 'File' menu, choose 'Open..' and select your input file (see below for the file's format). You can also drag your input file and drop it on the program's icon. After a short delay, your file should appear in its own scrollable and resizable window, to show you what the program has read. You can then be sure that the program read your data file correctly. NoteThe program has no way of knowing if your data file is corrupt. You should at least make a visual check on your data to ensure it has been read correctly.About threads..
Starting with version alpha 2, the program is multi-threaded, allowing the user to run multiple, concurrent analyses and background tasks. This requires the Thread Manager, part of System 7.5 and later. You can also get an extension for earlier systems. The 68k and PPC versions will be multi-threaded if the Thread Manager is present.When at least one data file has been read, the File menu's 'New analysis' item is enabled. Choosing it will cause the 'Multiple regression' window to come to the front of the screen. If you have opened more than one data window, the multiple regression will be carried out for the topmost data file. The multiple regression window looks like this: The names of the variables will be different for your own data file. The features of this window are as follows:
- A vector of values, with no matrix-like (distance) structure embedded: select 'Vector'. The values of the dependent variable will be permuted totally at random. This option may be used, for instance
- to test the behavior of the permutation tests of significance against standard multiple regression software, all possible permutations being equally likely, or
- to test multiple regression done on variables that do not meet the distributional assumptions of the standard parametric tests used in multiple regression, or else
- when your variables have no particular structure to them, e.g. they are simply a list of n values measured at n stations.
- A simple distance matrix (not a dendrogram or phylogenetic tree): select 'Matrix'. The rows and the columns of the original dependent distance matrix (Y) will be permuted at random in the manner of the Mantel test.
- A dendrogram (ultrametric tree): select 'Double' (Lapointe and Legendre, 1991).
- A phylogenetic (additive) tree: select 'Triple' (Lapointe and Legendre, 1992). In this case, you will need to indicate which object is the 'Rooting Object' in the popup menu. This is necessary in order for the program to extract the star tree that forms the additive component of the phylogenetic tree.
- The first one, dubbed 'Simple test', computes the regression coefficients (b's) of the selected variables (see below), permutes the data according to the permutation method you selected, and re-computes the regression coefficients k times, where k is the number of permutations.
- Backward elimination procedure
With this procedure, all the user-selected variables (see below) are initially included. The permutation test is conducted, and at each step the independent variable whose partial regression coefficient has the highest p-value is dropped, provided that this probability is also higher than a predetermined and Bonferroni-corrected p-to-remove value, which the user must select from the popup menu - Forward selection procedure
This procedure is based on the fact that a variable should be included in the model if (a) it gives the equation with the most significant R2 (Sokal and Rohlf, 1981) and (b) its b coefficient is significant at the Bonferroni-corrected p-to-enter level.
A forward selection procedure can be defined as follows: at each step, the independent matrix-variable is entered that produces the equation whose R2 coefficient has the lowest probability, provided that this probability is also smaller than or equal to a predetermined Bonferroni-corrected p-to-enter value. This variable's b coefficient must also be significant at the p-to-enter level, again pending approval through Bonferroni's correction. If a tie occurs in the permutational probabilities, the value of the increment in R2 is used as an additional selection criterion. Notice that we cannot use here the forward selection criterion based on the variable with the highest partial correlation, which is computationally simpler and is, in ordinary regression, equivalent to the one used here, because the relation between the values of the partial correlations and their associated probabilities may not hold, due to the special ways the permutations are performed in matrix regression.
The user must also select a p-to-enter value. - Stepwise regression
This procedure is a combination of the backward elimination and forward selection procedures. It consists in a forward selection where each forward step (at the p-to-enter level) is followed by a backward elimination (at the p-to-remove level), assessing the significance of all variables already entered in the model.
Unfortunately, stepwise regression is broken in the current version of Permute!, and the author does not have enough free time to go back and fix it. However, a forward selection followed by a backward elimination of the selected variables should provide a reasonable facsimile of a stepwise regression.
Test results
Test results are shown in a scrolling window. Currently, the program does not respond to user input while it is performing the computations. This will change in future revisions, but for now it means that exploratory analyses should be done with a limited number of permutations (e.g. 99 or 49, even 24 if the dataset is large). A good indication of the time needed to compute one permutation is to press the 'Quick test' button and see how much time is necessary for the results to appear. Multiply that time by the number of permutations and you have an indication of the total time needed before results start to appear. The 'Test results' window may be scrolled back and resized as desired. A run may have as many windows as the computer memory allows. You can copy the results to the clipboard (Edit menu.. Copy), save them to a file (File menu.. Save) and print them. There is a limit of 32,000 characters on the results window, the earlier characters being flushed as needed. In practice this limit is unlikely to be encountered, but the author is aware of the situation.Input file format
Although the computations are based on distance matrices, the program uses a clever scheme to simulate matrix-like behavior in vectors, in order to maximize memory usage, using some properties of symmetric matrices. If 'A' is a distance matrix, and i and j are integers, the following will be true: Therefore, the only unique part (i.e., that cannot be deduced from information found elsewhere) in the matrix is the upper-triangular part, excluding the diagonal. Of course, the lower-triangular could have been used instead; we selected the upper-triangular arbitrarily.Here is an illustration of the way the original distance matrices should be 'unfolded': The input file thus consist of the 'unfolded' upper-triangular matrices, read by rows, in a vertical format similar to the one found in many spreadsheet applications. There are as many columns to the file as necessary, each column representing an unfolded, upper-triangular matrix-variable. Columns should be separated by tabs. The first line of the file should be the name of the matrix-variables (separated by tabs); up to 10 characters per variable name will be read by the program. Following are a series of distance matrices, followed by the corresponding input files. The input files consist of the 'unfolded' upper triangular matrices (boldface values), excluding the diagonal. For each matrix, there is only one value per row in the input file.
Permute 3 4 4 Equals 1/3
File name: MyInputFile.txt File contents:
Technical notes
Limitations removed in version 3.4
The previous version of the program required two files (one for the Y matrix-variable and one for the X1.Xn matrix-variables), and the order in which the X appeared in the file determined their names in the program. For instance, the third column of the file containing the X matrix-variables was always named 'X3'. Furthermore, there were several limitations in the previous version of the program, such as:- Maximum of 6 independent matrix-variables
- The number of lines in each file had to be an integer that satisfied the n(n-1)/2 relationship: you could have 105 lines in your input file (meaning 15 objects in the matrix: 15*14/2 = 105) but not 104 or 106, as these numbers do not correspond to an integer number of objects and hence, the data file can't be a distance matrix..
- Maximum of 29 objects (406 lines) in all input files
Current limitations
- Maximum number of rows in a data file: 178,956,970
(CodeWarrior seems to accept a maximum size of MaxLongInt, or 2,147,483,647, but this is the highest Think Pascal allows me to go without complaining that a 'variable of this type would be too large'.. - Maximum number of columns (variables) in a data file: 178,956,970
- Note: due to limitations in the List Manager if your matrix has more than 16,000 cells or so it will show up as blank. It will be used in its entirety for the computations, though.
Requirements
Permute 3 4 4 Equals 2/3
Any Macintosh capable of running System 7.0 and up should be able to run Permute! 3.4, which has been tested on the following machines:Computer | Configuration | Processor | Time required |
Macintosh Plus | 4MB RAM, System 7.0.1 | 68000@16MHz | [Macro error: Can't call the script because the name 'elapsedTime' hasn't been defined.] |
Macintosh II | 8MB RAM, System 7.5.5 | 68020+FPU@16MHz | [Macro error: Can't call the script because the name 'elapsedTime' hasn't been defined.] |
Macintosh II | 8MB RAM, System 7.5.5 | 68020@16MHz (no FPU) | [Macro error: Can't call the script because the name 'elapsedTime' hasn't been defined.] |
Macintosh SE/30 | 20MB RAM, System 7.5.5 | 68030+FPU@16MHz | |
Macintosh IIci | 20MB RAM, System 7.5.5 | 68030+FPU@25MHz | [Macro error: Can't call the script because the name 'elapsedTime' hasn't been defined.] |
PowerBook 160 | 14MB RAM, System 7.5.3 | 68030@25MHz | |
Macintosh Classic II | 4MB RAM, System 7.5.5 | 68030@16MHz | [Macro error: Can't call the script because the name 'elapsedTime' hasn't been defined.] |
Power Macintosh 7100/66 | 48MB RAM, System 7.5.5 | PPC 601@66MHz (256k L2 Cache) | [Macro error: Can't call the script because the name 'elapsedTime' hasn't been defined.] |
Power Macintosh 7200/90 | 64MB RAM, System 8.0 | PPC 601@90MHz | [Macro error: Can't call the script because the name 'elapsedTime' hasn't been defined.] |
Motorola StarMax 180 | 80MB RAM, System 7.6.1 | PPC 603e@180MHz | [Macro error: Can't call the script because the name 'elapsedTime' hasn't been defined.] |
Power Macintosh 9600/350 | 64MB RAM, System 7.6.1 | PPC 604e@350MHz | [Macro error: Can't call the script because the name 'elapsedTime' hasn't been defined.] |
References
Hope, A. C. A. 1968. A simplified Monte Carlo significance test procedure. J. Roy. Stat. Soc. Ser. B 30: 582-598. Lapointe, F.-J., and P. Legendre. 1991. The generation of random ultrametric matrices representing dendrograms. J. Class. 8: 177-200.
Lapointe, F.-J. & P. Legendre. 1992. A statistical framework to test the consensus among additive trees (cladograms). Syst. Biol.41: 158-171.
Legendre, P., F.-J. Lapointe & P. Casgrain. Yate 4 7 1 1 3 answer. 1994. Modeling brain evolution from behavior: a permutational regression approach. Evolution48: 1487-1499.
Sokal, R.R. and F.J. Rohlf. 1981. Biometry, second edition. W.H. Freeman and Co., San Francisco. 859 pp.