4.4.4 Counting - Permutations Rule with Some Identical Items

Sasha Townsend - Tulsa

11 Oct 202020:46

EducationalLearning

32 Likes 10 Comments

TLDRThis video script explores permutations with identical items, explaining how to calculate the number of unique arrangements when some elements are the same. It introduces a formula involving factorials and demonstrates its application with examples, including survey design and arranging letters in 'Mississippi'. The intuitive reasoning behind the formula is clarified, showing how to avoid overcounting identical items in various scenarios.

Takeaways

📚 The lesson discusses permutations with identical items, explaining how to calculate the number of different arrangements when some items are the same.
🔢 The formula for permutations with identical items is given as n! divided by the product of factorials of the counts of each identical group (n1! * n2! * ... * nk!).
📉 The factorial rule is modified to account for identical items, preventing overcounting of arrangements that are essentially the same due to indistinguishable elements.
📝 An example of designing surveys is used to illustrate the concept, where repeating questions helps identify thoughtless or dishonest responses.
🔑 The reasoning behind the formula is explained through the example of arranging survey questions with identical questions to avoid overcounting.
🧩 The example of arranging survey questions with 10 total, 2 identical pairs, and 3 identical questions results in 302,400 different arrangements.
🌐 Another example involves arranging the letters of 'mississippi', which has repetitions of certain letters, leading to fewer unique arrangements than if all letters were unique.
🤔 The concept of overcounting is emphasized, explaining why we divide by the factorial of the number of identical items to correct for indistinguishable arrangements.
📈 The calculation for 'mississippi' shows that with 11 letters, 4 's's, 4 'i's, and 2 'p's, there are 34,650 unique arrangements.
🔠 The lesson concludes by highlighting the importance of adjusting for identical elements in permutations to get accurate counts of unique arrangements.
👋 The instructor signs off, indicating the next lesson will cover combinations, suggesting a continuation of the combinatorial mathematics theme.

Q & A

What is the main topic discussed in the video?
-The main topic discussed in the video is permutations with identical elements, where the number of arrangements is calculated differently when some items are the same.
What is the formula for calculating permutations when some items are identical?
-The formula for calculating permutations with identical items is to take the factorial of the total number of items (n!) and then divide by the factorial of the counts of each identical group (n1! * n2! * ... * nk!).
Can you provide an example from the video where the formula is applied?
-An example from the video is a survey with 10 questions, where 2 questions are identical to each other and 3 questions are identical to each other. Using the formula, the number of different arrangements possible is calculated as 10! / (2! * 3!).
What is the intuitive reason for the formula mentioned in the video?
-The intuitive reason for the formula is that it modifies the factorial rule to account for identical items, ensuring that arrangements that are the same due to identical items are not overcounted.
How does the video explain the concept of overcounting in permutations with identical items?
-The video explains overcounting by showing that if identical items are treated as unique during the factorial calculation, the same arrangement can be counted multiple times, which is not desired.
What is the significance of the survey example in the video?
-The survey example demonstrates how pollsters might use permutations with identical items to ensure that repeated questions in a survey do not lead to overcounting of identical arrangements.
How many different arrangements are possible for the survey example discussed in the video?
-For the survey example with 10 questions, where 2 questions are identical and 3 questions are identical, the number of different arrangements is 302,400.
What is another example used in the video to illustrate the concept of permutations with identical items?
-Another example used in the video is the arrangement of the letters in the word 'mississippi', where some letters are repeated.
How many arrangements of the letters in 'mississippi' are possible according to the video?
-According to the video, there are 34,650 different arrangements possible for the letters in 'mississippi', considering the repetitions of certain letters.
What is the role of factorials in the formula for permutations with identical items?
-Factorials play a crucial role in the formula as they represent the total number of ways to arrange items. Dividing by the factorial of the counts of identical items corrects for overcounting in permutations.
Why might dividing by 1! not change anything in the formula?
-Dividing by 1! does not change anything because 1! equals 1, so dividing by it is effectively the same as multiplying by 1, which does not alter the original number.

Outlines

00:00

🔢 Understanding Permutations with Identical Items

This paragraph introduces the concept of permutations where some items are identical. It explains how to calculate the number of unique permutations when selecting 'n' items with replacement, considering that some items may repeat (n1, n2, ..., nk). The formula involves taking the factorial of the total items and dividing by the factorial of the counts of each set of identical items. An example of designing surveys with repeated questions to detect thoughtless or dishonest answers is provided to illustrate the concept.

05:01

📝 Applying Permutation Formula to Survey Design

The paragraph delves into applying the permutation formula to a specific example of survey design. It outlines a scenario where a survey with 10 questions has two identical pairs and one identical triplet of questions. The process of calculating the number of unique arrangements by using the permutation formula is explained step by step, emphasizing the division by the factorial of the number of identical items to avoid overcounting.

10:03

🧩 Counting Arrangements of the Word 'Mississippi'

This section discusses a classic combinatorics problem of arranging the letters of the word 'Mississippi'. It highlights the presence of repeated letters and explains the necessity of using the permutation formula to account for identical letters. The calculation involves identifying the number of each letter and dividing the total factorial by the factorial of the counts of each set of identical letters, resulting in a reduced number of unique arrangements compared to a scenario with all unique letters.

15:04

📚 Detailed Explanation of Permutations with Identical Letters

The paragraph provides a detailed explanation of how to handle permutations with identical elements, using the example of the word 'Mississippi'. It breaks down the process of counting unique arrangements by considering the repeated letters (four 's's, four 'i's, and two 'p's) and dividing the total factorial by the factorial of the counts of each set of identical letters. This approach ensures that identical arrangements are not overcounted.

20:05

🚀 Conclusion on Permutations with Identical Elements

The final paragraph wraps up the lesson on permutations with identical elements. It summarizes the importance of using the correct formula to account for overcounting due to identical items. The paragraph concludes by setting the stage for the next lesson, which will cover combinations, indicating a continuation of the combinatorial topic.

Mindmap

Keywords

💡Permutations

Permutations refer to the various ways in which a set of items can be arranged in a specific order, where the order is significant. In the video, permutations are discussed in the context of selecting items with some being identical to each other. The formula for calculating permutations when items are identical is given as dividing the total number of items factorial by the factorial of the counts of each identical group, which is essential for understanding the arrangements of items where some are not unique.

💡Identical Items

Identical items are items that are the same and indistinguishable from one another. The video script explains that when calculating permutations, identical items must be accounted for differently because they do not create unique arrangements when swapped. This is illustrated with the survey example where two questions are identical, and the arrangement of these questions does not create a new unique survey.

💡Factorial

The factorial of a number, denoted as n!, is the product of all positive integers less than or equal to n. In the video, the concept of factorial is used to calculate the total number of permutations when initially considering all items as unique. The script then adjusts this by dividing by the factorial of the counts of identical items to avoid overcounting arrangements that are essentially the same.

💡Survey Design

Survey design is the process of creating a questionnaire to collect data from respondents. The video uses the example of survey design to illustrate the concept of permutations with identical items. Pollsters may repeat questions to check for consistency in responses, and the arrangement of these questions can be calculated using the formula for permutations with identical items.

💡Mississippi

In the video, the word 'Mississippi' is used as an example to demonstrate how to calculate permutations with identical letters. The word has 11 letters with repetitions of 's', 'i', and 'p', which means not all letter arrangements are unique. The script calculates the number of unique arrangements of the letters in 'Mississippi' using the permutations formula for identical items.

💡Combinatorics

Combinatorics is a branch of mathematics concerned with counting, combination, and permutation of sets. The video script delves into combinatorics when explaining how to calculate permutations with identical items, which is a fundamental concept in this field. The word 'combinatorics' is mentioned in the context of a classic counting problem involving the word 'Mississippi'.

💡Overcounting

Overcounting occurs when the same outcome is counted multiple times in a calculation. The video script discusses overcounting in the context of permutations with identical items, where initially, all items are treated as unique, leading to multiple identical arrangements being counted as different. The formula provided corrects for overcounting by dividing by the factorial of the counts of each group of identical items.

💡Rearrange

To rearrange means to put items in a different order. In the video, the concept of rearranging is central to the discussion of permutations, especially when some items are identical. The script uses the rearrangement of letters in 'Mississippi' to demonstrate how the formula for permutations with identical items works.

💡Divide by n_sub_k factorial

This phrase refers to the step in the formula for calculating permutations with identical items where you divide by the factorial of the number of identical items in each group (n_sub_k). The video script explains that this step is crucial for adjusting the total permutations to account for identical items, ensuring that arrangements that are essentially the same are not overcounted.

💡n_sub_k

In the context of the video, n_sub_k represents the number of identical items in a group. The subscript k distinguishes between different groups of identical items, such as n1, n2, ..., nk. The script uses n_sub_k in the formula for permutations with identical items to indicate that for each group of identical items, you must divide by the factorial of the number of items in that group.

Highlights

The video discusses learning outcome number four from lesson 4.4, focusing on permutations with identical items.

A formula is presented for calculating permutations when some items are identical, involving factorials and divisions by the factorial of the counts of identical items.

An intuitive explanation is given for the formula, relating it to the factorial rule and accounting for identical items.

A real-world example of designing surveys is used to illustrate the concept of identical questions and their permutations.

The survey example involves 10 questions with two identical pairs and a triplet, demonstrating the application of the permutation formula.

The process of reasoning through the survey example is shown step by step, aligning with the formula's calculation.

The concept of overcounting identical arrangements is explained, and the need to divide by the factorial of identical items is emphasized.

A detailed breakdown of the survey example's calculation is provided, showing the division by 2 factorial and 3 factorial.

The result of the survey example calculation is given as 302,400 different arrangements.

A classic combinatorics problem involving the word 'mississippi' is introduced to further explain the permutation formula.

The 'mississippi' problem highlights the presence of repeated letters and the need to account for them in permutation calculations.

The number of letters in 'mississippi' is counted, and the presence of identical letters is noted for the permutation calculation.

A step-by-step approach to calculating permutations for 'mississippi' is detailed, including the division by factorials of identical letters.

The final calculation for 'mississippi' results in 34,650 arrangements, showcasing the impact of identical letters on permutation count.

The lesson concludes with a summary of the permutation formula for identical elements and its practical applications.

Transcripts

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