Mathematics · Class 11

Active learning ideas

Permutations with Specific Conditions

Let's move beyond simple line-ups and explore more challenging arrangement puzzles. How many unique ways can you arrange the letters in a word like 'SUCCESS', or seat your friends around a campfire?

• CBSE Learning Outcomes
• NCERT Class 11: Chapter 7 - Permutations and Combinations

• 15–20 min
• Pairs → Whole Class
• 3 activities

Activity 01

• Collaborative Problem-Solving
• 20 min · Small Groups

Word Scramble Challenge

Give students sets of letter tiles for words with repeated letters like 'COMMITTEE' or 'MISSISSIPPI'. In small groups, they first find the total arrangements as if all letters were unique, then physically rearrange the identical letters to see that the word doesn't change, leading them to the idea of division.

Explain why we divide by the factorial of the count of repeated items when arranging non-distinct objects.

Facilitation Tip

Encourage groups to write down all the 'new' words they can make and then discuss why many look the same.

What to look for

Give an exit ticket with two questions: one on arranging the letters of 'ENGINEERING' and another on seating 5 people at a round table. This quickly checks understanding of the two main concepts.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 02

• Collaborative Problem-Solving
• 15 min · Whole Class

Human Line vs. Round Table

Ask 5 students to come to the front. First, calculate the number of ways they can stand in a line. Then, have them form a circle and discuss how many arrangements are truly different, guiding them to discover the (n-1)! formula for circular permutations.

Analyse how to calculate the number of arrangements of the letters in 'MISSISSIPPI'.

Facilitation Tip

Physically rotate the circle of students to show that it's the same relative arrangement until someone new is on the left or right.

What to look for

A section in a unit test containing multi-step word problems, such as: 'Find the number of ways 6 boys and 5 girls can be arranged in a line so that no two girls are together'.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 03

• Collaborative Problem-Solving
• 15 min · Pairs

Conditional Seating Chart

Present a problem on the board: 'In how many ways can 4 boys and 3 girls be seated in a row such that all 3 girls sit together?'. Students work in pairs using placeholders (like B1, B2, G1) to model the problem, treating the 'girls block' as one unit.

Compare the number of linear arrangements of 5 people with the number of circular arrangements of 5 people.

Facilitation Tip

After they solve for 'together', challenge them to find the number of ways the girls 'never' sit together.

What to look for

Provide a worksheet with a mix of problems (identical items, circular, restrictions) and a detailed answer key. Students can attempt the problems and then check their work to identify their own strengths and weaknesses.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

A few notes on teaching this unit

Begin by using physical letter tiles to demonstrate why we divide for repeated items, making the abstract formula tangible. For restrictions, use the analogy of 'tying' objects together with a string to form a single block. Then, introduce the 'gap method' as a powerful strategy for keeping objects separate.

Students will master the techniques to solve complex permutation problems involving repeated items, items that must stay together or apart, and circular arrangements.

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Watch Out for These Misconceptions

• Students use n! for circular permutations instead of (n-1)!

  Explain that in a linear arrangement, there are two ends which act as fixed reference points. In a circle, there are no ends. We must first 'fix' one person's position to create a reference point, and then we can arrange the remaining (n-1) people in (n-1)! ways.

• For 'never together' problems, students calculate arrangements for each object separately and subtract from the total.

  This is incorrect as it leads to double counting. The correct method is to calculate the total number of unrestricted arrangements and subtract the number of arrangements where the objects are 'always together'. Total Arrangements - Always Together Arrangements = Never Together Arrangements.

• Students forget to multiply by the internal arrangement of the 'grouped' items.

  When using the 'string method' where several items are treated as one unit (e.g., all girls sit together), remind students that the items within that unit can also be arranged among themselves. They must multiply the factorial of the main arrangement by the factorial of the internal arrangement.

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Methods used in this brief

• Collaborative Problem-Solving