Lesson Note on Permutations and Combinations: Meaning and Types of Permutations for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS2)
Term: 3rd Term
Week: 4
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Combinatorics
Previous Lesson: Trigonometric Function: Equations and Identities.
Topic: PERMUTATIONS AND COMBINATIONS
Subject Matter: Meaning of permutations and combinations (counting arrangements and selections), permutation on arrangement, cyclic permutation, arrangement of identical objects.

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

• Define permutation and combination.
• Differentiate between permutation and combination.
• State the formulas for linear, cyclic, and identical permutations.
• Solve problems involving various types of permutations.

Affective Domain:

• Appreciate the application of permutations in real-life scenarios.
• Develop interest in solving problems related to combinatorics.

Psychomotor Domain:

• Accurately apply permutation formulas to solve given problems.
• Present clear, step-by-step solutions to permutation questions.

Social Domain:

• Collaborate with peers to discuss and solve permutation problems.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• Further Mathematics for Senior Secondary Schools by A.O. Salami et al.
• https://www.examguides.com
• https://www.mathsisfun.com/combinatorics/combinations-permutations.html

Instructional Materials

The teacher will teach this lesson with the aid of:

• Charts showing functional notation (n!), nPr, and nCr formulas.
• Whiteboard and markers.
• Calculators.

Rationale for the Lesson

This lesson helps pupils understand how to count arrangements and selections of objects, which is important for solving problems in probability, statistics, and various real-life situations like scheduling or coding. It builds a foundation for more advanced mathematical concepts.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of basic arithmetic operations, factorials, and simple counting principles.

Lesson Content/Board Summary

PERMUTATIONS AND COMBINATIONS

Meaning of Permutations and Combinations

Permutation refers to the arrangement of objects in a specific order. The order of arrangement matters in permutations.

Combination refers to the selection of objects where the order of selection does not matter. It is a group or subset of objects.

Permutations (Arrangement of Distinct Objects)

A permutation is an ordered arrangement of objects. The number of permutations of ‘n’ distinct objects taken ‘r’ at a time is given by the formula:

P(n, r) = nPr = n! / (n – r)!

Where n! (n factorial) = n × (n-1) × (n-2) × … × 1.

Example 1: How many ways can the first, second, and third prizes be awarded to 8 contestants?

Solution:

Step 1: Identify n and r. Here, n = 8 (total contestants), r = 3 (prizes).

Step 2: Apply the permutation formula: P(8, 3) = 8! / (8 – 3)!

Step 3: Calculate the factorial: P(8, 3) = 8! / 5! = (8 × 7 × 6 × 5!) / 5!

Step 4: Simplify: P(8, 3) = 8 × 7 × 6 = 336 ways.

Example 2: In how many ways can 5 different books be arranged on a shelf?

Solution:

Step 1: Identify n and r. Here, n = 5 (total books), r = 5 (all books are arranged).

Step 2: Apply the permutation formula: P(5, 5) = 5! / (5 – 5)! = 5! / 0!

Note: 0! = 1.

Step 3: Calculate the factorial: P(5, 5) = 5 × 4 × 3 × 2 × 1 = 120 ways.

Cyclic Permutations

Cyclic permutation is the arrangement of objects in a circle. If ‘n’ distinct objects are arranged in a circle, the number of cyclic permutations is given by:

(n – 1)!

Example 1: In how many ways can 7 people be seated around a circular table?

Solution:

Step 1: Identify n. Here, n = 7 (number of people).

Step 2: Apply the cyclic permutation formula: (n – 1)! = (7 – 1)!

Step 3: Calculate the factorial: 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 ways.

Example 2: How many different ways can 4 beads of different colours be arranged to form a necklace?

Solution:

Step 1: Identify n. Here, n = 4 (number of beads).

Step 2: Apply the cyclic permutation formula: (n – 1)! = (4 – 1)!

Step 3: Calculate the factorial: 3! = 3 × 2 × 1 = 6 ways.

Permutations with Identical Objects

When some of the objects are identical, the number of distinct permutations of ‘n’ objects where p objects are of one type, q objects are of a second type, r objects are of a third type, and so on, is given by:

n! / (p! × q! × r! …)

Example 1: Find the number of distinct permutations of the letters in the word “MATHEMATICS”.

Solution:

Step 1: Count the total number of letters (n). n = 11.

Step 2: Identify identical letters and their counts:

• M appears 2 times (p=2)
• A appears 2 times (q=2)
• T appears 2 times (r=2)
• H, E, I, C, S appear 1 time each.

Step 3: Apply the formula: 11! / (2! × 2! × 2!)

Step 4: Calculate: 39,916,800 / (2 × 2 × 2) = 39,916,800 / 8 = 4,989,600 distinct permutations.

Example 2: How many distinct permutations can be formed from the letters of the word “APPLE”?

Solution:

Step 1: Count the total number of letters (n). n = 5.

Step 2: Identify identical letters and their counts:

• P appears 2 times (p=2)
• A, L, E appear 1 time each.

Step 3: Apply the formula: 5! / 2!

Step 4: Calculate: (5 × 4 × 3 × 2 × 1) / (2 × 1) = 120 / 2 = 60 distinct permutations.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Set Induction

Teacher’s Activity: The teacher asks pupils how many different ways they can arrange 3 different books on a shelf to introduce the concept of arrangement.

Pupils’ Activity: Pupils attempt to provide different arrangements and share their ideas.

Learning Point: Pupils recall prior knowledge of counting and arrangement, leading to the lesson topic.

Step 2: Meaning of Permutations and Combinations

Time: 7 minutes

Teaching Skill: Explanation/Definition

Teacher’s Activity: The teacher defines permutation and combination, highlighting the key difference that order matters in permutations but not in combinations. The teacher uses simple examples like arranging letters vs. selecting a committee.

Pupils’ Activity: Pupils listen attentively, ask questions for clarity, and take notes on the definitions.

Learning Point: Pupils understand the fundamental difference between permutations and combinations.

Step 3: Linear Permutations

Time: 10 minutes

Teaching Skill: Formula Derivation/Demonstration

Teacher’s Activity: The teacher introduces the formula for linear permutations (nPr) and explains each variable. The teacher solves the first set of worked examples from the board summary, explaining each step clearly.

Pupils’ Activity: Pupils copy the formula and worked examples, asking questions about the steps and calculations.

Learning Point: Pupils learn how to apply the linear permutation formula to solve problems.

Step 4: Cyclic Permutations

Time: 7 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher explains the concept of cyclic permutations and provides the formula (n-1)!. The teacher solves the worked examples for cyclic permutations from the board summary.

Pupils’ Activity: Pupils follow the explanation, copy the formula and examples, and attempt to solve similar problems mentally.

Learning Point: Pupils understand how to calculate permutations in a circular arrangement.

Step 5: Permutations with Identical Objects

Time: 6 minutes

Teaching Skill: Explanation/Problem Solving

Teacher’s Activity: The teacher explains how to handle permutations when identical objects are present, providing the formula n! / (p!q!r!…). The teacher solves the worked examples for identical objects from the board summary.

Pupils’ Activity: Pupils listen, copy the formula and examples, and clarify any doubts regarding repeated objects.

Learning Point: Pupils learn to calculate permutations when some objects are indistinguishable.

Step 6: Further Worked Examples/Practice

Time: 5 minutes

Teaching Skill: Guided Practice

Teacher’s Activity: The teacher provides additional problems covering all types of permutations discussed and guides pupils to solve them on the board or in groups.

Pupils’ Activity: Pupils actively participate in solving the problems, discussing methods, and seeking assistance when needed.

Learning Point: Pupils consolidate their understanding and improve their problem-solving skills.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Define permutation and combination.
2. State the formula for calculating permutations of ‘n’ distinct objects taken ‘r’ at a time.
3. How many ways can 6 people be arranged in a line?
4. How many distinct permutations can be formed from the letters of the word “BANANA”?
5. In how many ways can 5 keys be arranged on a key ring?

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 5 minutes

Teaching Skill: Summarization

Teacher’s Activity: The teacher summarizes the key learning points of the lesson, reiterating the definitions and formulas for different types of permutations. The teacher then assigns homework from the textbook.

Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.

Learning Point: Pupils recall the main concepts and consolidate their learning.

Lesson Keywords

• Permutation – An arrangement of objects in a specific order.
• Combination – A selection of objects where order does not matter.
• Factorial – The product of all positive integers less than or equal to a given positive integer (e.g., 5! = 5x4x3x2x1).
• Arrangement – The way in which things are placed or organized.
• Cyclic Permutation – The arrangement of objects in a circle.
• Identical Objects – Objects that are indistinguishable from one another.

Differentiation

For struggling learners, the teacher will provide simpler problems with smaller numbers and offer one-on-one guidance. Advanced learners will be challenged with more complex problems involving multiple conditions or real-world application scenarios, encouraging them to explore combinations as well.

Note for teachers using this lesson plan

Ensure that pupils clearly understand the difference between permutation and combination before proceeding to the formulas. Emphasize the importance of identifying whether order matters in a given problem. Use visual aids like charts for formulas to reinforce learning. Encourage pupils to show all steps in their calculations to help them track their work and identify errors. Provide ample practice exercises to build confidence and proficiency.