This is one of the following two articles on Combinations and Permutations in Excel

Combinations in Excel 2010 and Excel 2013

Permutations in Excel 2010 and Excel 2013

# Permutations Overview

Both combinations and permutations represent all the possible ways to arrange k items that are randomly and repeatedly selected from a larger set of n items.

A single permutation describes a unique ordering of those k items. For example, the combination that contains the letters A, B, and C has six possible permutations (orderings) of that one combination (unordered collection) of letters. The six total different ways that the three letters A, B, and C can be ordered (ABC, ACB, BAC, BCA, CAB, CBA) represent the six total permutations of those three items (letters) that exist.

The total number of possible permutations equals the total number of possible combinations times the total number of permutations that can be derived from each combination. The preceding example shows that the set of four letters [ A, B, C, D ] has a total of four unique three-letter combinations. Six different permutations (orderings) can be derived from each one of these four combinations. The total number of three-letter permutations that can be created from the set of four letters [ A, B, C, D ] is 24.

A permutation can have the same items as another permutation but those items must be arranged in a different order to be considered a unique permutation.

Different arrangements or orderings of the same k items in a subset DO create different permutations. The number of possible permutations is the total number of ways that all of the possible subsets of k items can be ordered differently. The number of possible combinations is the total number of ways that k elements can be arranged from n total elements *when order of the k objects DOES matter*.

Re-arrangement of the same k elements within a subset creates a new permutation but does not create a new combination. There are always more permutations combinations of k objects taken from n total items because each combination of k elements can be re-ordered to create k! permutations. The total number of permutations of k items equals the following:

Total number of permutations of k items = (Total number of combinations of k items) * k!

##### Simultaneous vs. Sequential Selection

** Combinations** - Elements are picked simultaneously, all at once.

** Permutations** - Elements are picked sequentially, one after another, i.e., in a specific order.

If there is no order to the arrangement, it is a combination.

If there is any specified order to the arrangement, it is a permutation.

## Excel Functions For Permutations

The Number of Permutations equals the number of times that n different objects taken k at a time sequentially is given by the following:

**PERMUT ( n, k ) ****= Number of Permutations of n Objects Taken k at a time Sequentially**

**FACT ( n ) ****= n!**

----> FACT() is a Math & Trig Excel function, not a Statistical category function like PERMUT().

For example, the number of permutations of 9 objects taken 4 at a time sequentially =

= _{n}P_{k} = _{9}P_{4} = n! / (n - xk)!

= 9! / (9 - 4 )!

= FACT(n) / FACT(n - k)

= FACT(9) / FACT(5)

= 3,024

= PERMUT (n,k)

= PERMUT (9,4)

= 3,024

The Excel formulas for combinations and permutations are verified by referring back to the original example provided in this section as follows:

From the set of four letters [ A, B, C, D ], the following four unique three-letter combinations can be created:

A B C

A B D

D B A

D B A

This agrees with the result of the following Excel formula:

C(n,k) = COMBIN(n,k) = COMBIN(4,3) = 4

Six permutations can be derived from each one of the above three-letter combinations. The number of permutations that can be derived from a combination of k items equals k!. In this case k = 3. k! = 6.

The six different ways that the three letters A, B, and C can be ordered (ABC, ACB, BAC, BCA, CAB, CBA) represent six unique permutations of those three letters.

Total number of permutations of k items = (Total number of combinations of k items) * k!

Total number of permutations = 4 * 6 = 24

This agrees with the result of the following Excel formula:

P(n,k) = PERMUT(n,k) = PERMUT(4,3) = 24

C(n,k) = 4

P(n,k) = 24

k! = 6

## Permutation Problems in Excel:

### Problem 1: Permutations of Delivery Routes

A milkman makes 7 deliveries on his route. How many different sequences can he make to complete all 7 stops?

This is a **permutation** problem because the stops are done **sequentially**. After each stop, there is one less stop to make.

n = 7 = total number of objects initially available for inclusion in each permutation group

k = 7 = number of objects that will sequentially fill the permutation group

The number of permutations of n = 7 different stops taken k = 7 at a time sequentially equals:

_{n}P_{k} = _{7}P_{7} = PERMUT(n,k) = PERMUT(7,7) = 5,040

### Problem 2: Permutations of Seating Arrangements

How many ways can 5 people be seated on a sofa if only 3 seats are available and the 3 seats are filled sequentially by the available 5 people?

This is a **permutation** problem because the elements of the permutation group are filled **sequentially**. When each person is seated, there is one less person available to be seated.

n = 5 = total number of objects initially available for inclusion in each permutation group

k = 3 = number of objects that will sequentially fill the permutation group

The number of permutations of n = 5 different people seated k = 3 at a time sequentially equal:

_{n}P_{k} = _{5}P_{3} = PERMUT(n,k) = PERMUT(5,3) = 60

### Problem 3: Permutations of Executive Groups

A group of 9 people needs to appoint 1 person to be group president, another person to be group vice president, and a third person to be group treasurer. If the group first votes for the president, then votes for the vice president, and finally votes for the treasurer, how many different executive groups can be created from the original 9 people?

This is a **permutation** problem because the elements of the permutation group are filled up **sequentially**. When each position is filled, there is one less person available for a position.

n = 9 = total number of objects initially available for inclusion in each permutation group

k= 3 = number of objects that will sequentially fill the permutation group

The number of permutations of n = 9 different people elected k = 3 at a time sequentially equals:

_{n}P_{k} = _{9}P_{3} = PERMUT(n,k) = PERMUT(9,3) = 504

### Problem 4: Permutations of Book Arrangements

How many ways can 3 books be placed next to each other on a shelf one at a time?

This is a **permutation** problem because the elements of the permutation group are filled **sequentially**. When each book is placed, there is one less book to place.

n = 3 = total number of initially objects available for inclusion in each permutation group

k = 3 = number of objects that will sequentially fill the permutation group

The number of permutations of n = 3 different books placed k = 3 at a time sequentially equals:

_{n}P_{k} = _{3}P_{3} = PERMUT(n,k) = PERMUT(3,3) = 6

### Problem 5: Permutations of Letter Groups

From the following six letters: A, B, C, D, E, F, how many groups of 3 letters can be created if none of the letters from the original 6 are repeated in any group?

This is a **permutation** problem because **none of the letters can be repeated. **When the first letter of one of the permutation groups is chosen, there are only five remaining letters to choose from. Thus, the elements of the permutation group are filled **sequentially**.

n = 6 = total number of objects initially available for inclusion in each permutation group

k = 3 = number of objects that will sequentially fill the permutation group

The number of permutations of n = 6 different letters placed k = 3 at a time sequentially equally:

_{n}P_{k} = _{6}P_{3} = PERMUT(n,k) = PERMUT(6,3) = 120

**Excel Master Series Blog Directory**

Statistical Topics and Articles In Each Topic

- Histograms in Excel
- Bar Chart in Excel
- Combinations & Permutations in Excel
- Normal Distribution in Excel
- Overview of the Normal Distribution
- Normal Distribution’s PDF (Probability Density Function) in Excel 2010 and Excel 2013
- Normal Distribution’s CDF (Cumulative Distribution Function) in Excel 2010 and Excel 2013
- Solving Normal Distribution Problems in Excel 2010 and Excel 2013
- Overview of the Standard Normal Distribution in Excel 2010 and Excel 2013
- An Important Difference Between the t and Normal Distribution Graphs
- The Empirical Rule and Chebyshev’s Theorem in Excel – Calculating How Much Data Is a Certain Distance From the Mean
- Demonstrating the Central Limit Theorem In Excel 2010 and Excel 2013 In An Easy-To-Understand Way

- t-Distribution in Excel
- Binomial Distribution in Excel
- z-Tests in Excel
- Overview of Hypothesis Tests Using the Normal Distribution in Excel 2010 and Excel 2013
- One-Sample z-Test in 4 Steps in Excel 2010 and Excel 2013
- 2-Sample Unpooled z-Test in 4 Steps in Excel 2010 and Excel 2013
- Overview of the Paired (Two-Dependent-Sample) z-Test in 4 Steps in Excel 2010 and Excel 2013

- t-Tests in Excel
- Overview of t-Tests: Hypothesis Tests that Use the t-Distribution
- 1-Sample t-Tests in Excel
- 1-Sample t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Normality Testing For the 1-Sample t-Test in Excel 2010 and Excel 2013
- 1-Sample t-Test – Effect Size in Excel 2010 and Excel 2013
- 1-Sample t-Test Power With G*Power Utility
- Wilcoxon Signed-Rank Test in 8 Steps As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013
- Sign Test As a 1-Sample t-Test Alternative in Excel 2010 and Excel 2013

- 2-Independent-Sample Pooled t-Tests in Excel
- 2-Independent-Sample Pooled t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Variance Tests: Levene’s, Brown-Forsythe, and F Test For 2-Sample Pooled t-Test in Excel 2010 and Excel 2013
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro Wilk Tests For Two-Sample Pooled t-Test
- Two-Independent-Sample Pooled t-Test - All Excel Calculations
- 2- Sample Pooled t-Test – Effect Size in Excel 2010 and Excel 2013
- 2-Sample Pooled t-Test Power With G*Power Utility
- Mann-Whitney U Test in 12 Steps in Excel as 2-Sample Pooled t-Test Nonparametric Alternative in Excel 2010 and Excel 2013
- 2- Sample Pooled t-Test = Single-Factor ANOVA With 2 Sample Groups

- 2-Independent-Sample Unpooled t-Tests in Excel
- 2-Independent-Sample Unpooled t-Test in 4 Steps in Excel 2010 and Excel 2013
- Variance Tests: Levene’s Test, Brown-Forsythe Test, and F-Test in Excel For 2-Sample Unpooled t-Test
- Excel Normality Tests Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk For 2-Sample Unpooled t-Test
- 2-Sample Unpooled t-Test Excel Calculations, Formulas, and Tools
- Effect Size for a 2-Independent-Sample Unpooled t-Test in Excel 2010 and Excel 2013
- Test Power of a 2-Independent Sample Unpooled t-Test With G-Power Utility

- Paired (2-Sample Dependent) t-Tests in Excel
- Paired t-Test in 4 Steps in Excel 2010 and Excel 2013
- Excel Normality Testing of Paired t-Test Data
- Paired t-Test Excel Calculations, Formulas, and Tools
- Paired t-Test – Effect Size in Excel 2010, and Excel 2013
- Paired t-Test – Test Power With G-Power Utility
- Wilcoxon Signed-Rank Test in 8 Steps As a Paired t-Test Alternative
- Sign Test in Excel As A Paired t-Test Alternative

- Hypothesis Tests of Proportion in Excel
- Hypothesis Tests of Proportion Overview (Hypothesis Testing On Binomial Data)
- 1-Sample Hypothesis Test of Proportion in 4 Steps in Excel 2010 and Excel 2013
- 2-Sample Pooled Hypothesis Test of Proportion in 4 Steps in Excel 2010 and Excel 2013
- How To Build a Much More Useful Split-Tester in Excel Than Google's Website Optimizer

- Chi-Square Independence Tests in Excel
- Chi-Square Goodness-Of-Fit Tests in Excel
- F Tests in Excel
- Correlation in Excel
- Pearson Correlation in Excel
- Spearman Correlation in Excel
- Confidence Intervals in Excel
- z-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013
- t-Based Confidence Intervals of a Population Mean in 2 Steps in Excel 2010 and Excel 2013
- Minimum Sample Size to Limit the Size of a Confidence interval of a Population Mean
- Confidence Interval of Population Proportion in 2 Steps in Excel 2010 and Excel 2013
- Min Sample Size of Confidence Interval of Proportion in Excel 2010 and Excel 2013

- Simple Linear Regression in Excel
- Overview of Simple Linear Regression in Excel 2010 and Excel 2013
- Complete Simple Linear Regression Example in 7 Steps in Excel 2010 and Excel 2013
- Residual Evaluation For Simple Regression in 8 Steps in Excel 2010 and Excel 2013
- Residual Normality Tests in Excel – Kolmogorov-Smirnov Test, Anderson-Darling Test, and Shapiro-Wilk Test For Simple Linear Regression
- Evaluation of Simple Regression Output For Excel 2010 and Excel 2013
- All Calculations Performed By the Simple Regression Data Analysis Tool in Excel 2010 and Excel 2013
- Prediction Interval of Simple Regression in Excel 2010 and Excel 2013

- Multiple Linear Regression in Excel
- Basics of Multiple Regression in Excel 2010 and Excel 2013
- Complete Multiple Linear Regression Example in 6 Steps in Excel 2010 and Excel 2013
- Multiple Linear Regression’s Required Residual Assumptions
- Normality Testing of Residuals in Excel 2010 and Excel 2013
- Evaluating the Excel Output of Multiple Regression
- Estimating the Prediction Interval of Multiple Regression in Excel
- Regression - How To Do Conjoint Analysis Using Dummy Variable Regression in Excel

- Logistic Regression in Excel
- Logistic Regression Overview
- Logistic Regression in 6 Steps in Excel 2010 and Excel 2013
- R Square For Logistic Regression Overview
- Excel R Square Tests: Nagelkerke, Cox and Snell, and Log-Linear Ratio in Excel 2010 and Excel 2013
- Likelihood Ratio Is Better Than Wald Statistic To Determine if the Variable Coefficients Are Significant For Excel 2010 and Excel 2013
- Excel Classification Table: Logistic Regression’s Percentage Correct of Predicted Results in Excel 2010 and Excel 2013
- Hosmer- Lemeshow Test in Excel – Logistic Regression Goodness-of-Fit Test in Excel 2010 and Excel 2013

- Single-Factor ANOVA in Excel
- Overview of Single-Factor ANOVA
- Single-Factor ANOVA in 5 Steps in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For Each Single-Factor ANOVA Sample Group
- Kruskal-Wallis Test Alternative For Single Factor ANOVA in 7 Steps in Excel 2010 and Excel 2013
- Levene’s and Brown-Forsythe Tests in Excel For Single-Factor ANOVA Sample Group Variance Comparison
- Single-Factor ANOVA - All Excel Calculations
- Overview of Post-Hoc Testing For Single-Factor ANOVA
- Tukey-Kramer Post-Hoc Test in Excel For Single-Factor ANOVA
- Games-Howell Post-Hoc Test in Excel For Single-Factor ANOVA
- Overview of Effect Size For Single-Factor ANOVA
- ANOVA Effect Size Calculation Eta Squared in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Psi – RMSSE – in Excel 2010 and Excel 2013
- ANOVA Effect Size Calculation Omega Squared in Excel 2010 and Excel 2013
- Power of Single-Factor ANOVA Test Using Free Utility G*Power
- Welch’s ANOVA Test in 8 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar
- Brown-Forsythe F-Test in 4 Steps in Excel Substitute For Single-Factor ANOVA When Sample Variances Are Not Similar

- Two-Factor ANOVA With Replication in Excel
- Two-Factor ANOVA With Replication in 5 Steps in Excel 2010 and Excel 2013
- Variance Tests: Levene’s and Brown-Forsythe For 2-Factor ANOVA in Excel 2010 and Excel 2013
- Shapiro-Wilk Normality Test in Excel For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication Effect Size in Excel 2010 and Excel 2013
- Excel Post Hoc Tukey’s HSD Test For 2-Factor ANOVA With Replication
- 2-Factor ANOVA With Replication – Test Power With G-Power Utility
- Scheirer-Ray-Hare Test Alternative For 2-Factor ANOVA With Replication

- Two-Factor ANOVA Without Replication in Excel
- Randomized Block Design ANOVA in Excel
- Repeated-Measures ANOVA in Excel
- Single-Factor Repeated-Measures ANOVA in 4 Steps in Excel 2010 and Excel 2013
- Sphericity Testing in 9 Steps For Repeated Measures ANOVA in Excel 2010 and Excel 2013
- Effect Size For Repeated-Measures ANOVA in Excel 2010 and Excel 2013
- Friedman Test in 3 Steps For Repeated-Measures ANOVA in Excel 2010 and Excel 2013

- ANCOVA in Excel
- Normality Testing in Excel
- Creating a Box Plot in 8 Steps in Excel
- Creating a Normal Probability Plot With Adjustable Confidence Interval Bands in 9 Steps in Excel With Formulas and a Bar Chart
- Chi-Square Goodness-of-Fit Test For Normality in 9 Steps in Excel
- Kolmogorov-Smirnov, Anderson-Darling, and Shapiro-Wilk Normality Tests in Excel

- Nonparametric Testing in Excel
- Mann-Whitney U Test in 12 Steps in Excel
- Wilcoxon Signed-Rank Test in 8 Steps in Excel
- Sign Test in Excel
- Friedman Test in 3 Steps in Excel
- Scheirer-Ray-Hope Test in Excel
- Welch's ANOVA Test in 8 Steps Test in Excel
- Brown-Forsythe F Test in 4 Steps Test in Excel
- Levene's Test and Brown-Forsythe Variance Tests in Excel
- Chi-Square Independence Test in 7 Steps in Excel
- Chi-Square Goodness-of-Fit Tests in Excel
- Chi-Square Population Variance Test in Excel

- Post Hoc Testing in Excel
- Creating Interactive Graphs of Statistical Distributions in Excel
- Interactive Statistical Distribution Graph in Excel 2010 and Excel 2013
- Interactive Graph of the Normal Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Chi-Square Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution’s PDF in Excel 2010 and Excel 2013
- Interactive Graph of the t-Distribution’s CDF in Excel 2010 and Excel 2013
- Interactive Graph of the Binomial Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Exponential Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Beta Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Gamma Distribution in Excel 2010 and Excel 2013
- Interactive Graph of the Poisson Distribution in Excel 2010 and Excel 2013

- Solving Problems With Other Distributions in Excel
- Solving Uniform Distribution Problems in Excel 2010 and Excel 2013
- Solving Multinomial Distribution Problems in Excel 2010 and Excel 2013
- Solving Exponential Distribution Problems in Excel 2010 and Excel 2013
- Solving Beta Distribution Problems in Excel 2010 and Excel 2013
- Solving Gamma Distribution Problems in Excel 2010 and Excel 2013
- Solving Poisson Distribution Problems in Excel 2010 and Excel 2013

- Optimization With Excel Solver
- Maximizing Lead Generation With Excel Solver
- Minimizing Cutting Stock Waste With Excel Solver
- Optimal Investment Selection With Excel Solver
- Minimizing the Total Cost of Shipping From Multiple Points To Multiple Points With Excel Solver
- Knapsack Loading Problem in Excel Solver – Optimizing the Loading of a Limited Compartment
- Optimizing a Bond Portfolio With Excel Solver
- Travelling Salesman Problem in Excel Solver – Finding the Shortest Path To Reach All Customers

- Chi-Square Population Variance Test in Excel
- Analyzing Data With Pivot Tables
- SEO Functions in Excel
- Time Series Analysis in Excel
- VLOOKUP

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