Unlocking the Secrets of Distinguishable Permutations

Algebra 2 Grades High School 3:58 Video
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Lesson Description

Master the art of finding distinguishable permutations in words, accounting for repeated letters to avoid overcounting. Learn factorial simplification and apply it to real-world examples.

Video Resource

Distinguishable Permutations of Letters in a Word

Mario's Math Tutoring

Duration: 3:58
Watch on YouTube

Key Concepts

  • Factorials (n!)
  • Permutations
  • Distinguishable Permutations
  • Multiplicity of Repeated Elements

Learning Objectives

  • Calculate factorials and simplify factorial expressions.
  • Determine the number of distinguishable permutations of a word with repeated letters.
  • Apply the formula for distinguishable permutations to solve various problems.

Educator Instructions

  • Introduction (5 mins)
    Begin by reviewing the concept of permutations and factorials. Briefly explain the problem of overcounting when dealing with repeated elements in a set. Introduce the term 'distinguishable permutations.'
  • Video Presentation (10 mins)
    Play the video 'Distinguishable Permutations of Letters in a Word' by Mario's Math Tutoring. Instruct students to take notes on the formula and the examples provided.
  • Guided Practice (15 mins)
    Work through the examples from the video, pausing to ask clarifying questions and ensuring student understanding. Emphasize the importance of identifying repeated letters and adjusting the factorial calculation accordingly.
  • Independent Practice (15 mins)
    Provide students with additional word problems to solve independently. Circulate to provide assistance and answer questions.
  • Wrap-up and Review (5 mins)
    Summarize the key concepts and address any remaining questions. Preview upcoming topics related to permutations and combinations.

Interactive Exercises

  • Word Scramble Challenge
    Divide students into groups and provide each group with a word containing repeated letters. Challenge them to find the number of distinguishable permutations for that word.
  • Factorial Simplification Game
    Present students with complex factorial expressions and have them simplify them as quickly as possible. This can be done individually or in teams.

Discussion Questions

  • Why is it important to account for repeated letters when calculating permutations?
  • Explain how factorials are used to represent the number of ways to arrange distinct objects.
  • Can you think of real-world scenarios where distinguishable permutations would be useful?

Skills Developed

  • Algebraic Manipulation
  • Problem-Solving
  • Critical Thinking
  • Combinatorial Reasoning

Multiple Choice Questions

Question 1:

What is the value of 5! (5 factorial)?

Correct Answer: 120

Question 2:

How many distinguishable permutations are there in the word 'BANANA'?

Correct Answer: 60

Question 3:

What is the formula for distinguishable permutations of a word with 'n' letters, where a letter is repeated 'r' times?

Correct Answer: n! / r!

Question 4:

How many distinguishable permutations are there in the word 'STATISTICS'?

Correct Answer: 50400

Question 5:

Simplify the expression 8! / 6!

Correct Answer: 56

Question 6:

Which of the following scenarios requires the use of distinguishable permutations?

Correct Answer: Arranging the letters in the word 'MISSISSIPPI'.

Question 7:

If a word has 8 letters with one letter repeated 3 times and another letter repeated twice, how many distinguishable permutations are possible?

Correct Answer: 6720

Question 8:

What is the value of 0! (zero factorial)?

Correct Answer: 1

Question 9:

How many ways can you arrange the letters in the word 'RACECAR'?

Correct Answer: 210

Question 10:

In the formula for distinguishable permutations, what does dividing by the factorial of the repeated letter's count achieve?

Correct Answer: It eliminates duplicate arrangements.

Fill in the Blank Questions

Question 1:

The product of all positive integers less than or equal to a given positive integer is called a ________.

Correct Answer: factorial

Question 2:

A permutation where some elements are identical is called a ________ permutation.

Correct Answer: distinguishable

Question 3:

To find the number of distinguishable permutations of a word with repeated letters, you must ________ by the factorial of the count of each repeated letter.

Correct Answer: divide

Question 4:

The number of ways to arrange 'n' distinct objects is given by ________.

Correct Answer: n!

Question 5:

In the word 'SUCCESS', the letter 'S' is repeated ________ times.

Correct Answer: 3

Question 6:

If a word has 6 letters and one letter is repeated twice, the formula to find the number of distinguishable permutations is 6! / ________.

Correct Answer: 2!

Question 7:

The value of 1! is ________.

Correct Answer: 1

Question 8:

The number of distinguishable permutations of the word 'LEVEL' is ________.

Correct Answer: 30

Question 9:

Simplifying factorial expressions involves ________ common factors.

Correct Answer: canceling

Question 10:

When calculating distinguishable permutations, we divide by the factorial of the number of repetitions to avoid ________.

Correct Answer: overcounting

Educational Standards

A2.D.3:Use probability to evaluate outcomes of decisions. A2.A.2:Generate and evaluate equivalent algebraic expressions and equations using various strategies.

Teaching Materials

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