MathematicsChapter 6: Permutation and Combination
Restrictions in Arrangements
Typical restrictions: some people must stand together (tie them as one block, then arrange the block internally).
Learning objectives
- State the main ideas and notation used in restrictions in arrangements.
- Apply the rules of restrictions in arrangements in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
Typical restrictions: some people must stand together (tie them as one block, then arrange the block internally). Some must not be together (total minus together). First/last slot fixed. Vowels together in a word. Meaning: rebuild the problem so the product rule still applies. Always start from the unrestricted total if the paper is a 'not together' question.
Where / symbols
- RESTRICTIONS IN ARRANGEMENTS
- Typical restrictions: some people must stand together (tie them as one block, then arrange the block internally). Some must not be together (total minus together). First/last slot fixed. Vowels together in a word. Meaning: rebuild the problem so the product rule still applies. Always start from the unrestricted total if the paper is a 'not together' question.
- 1. TOGETHER
- n people, 2 must be together: treat the pair as one, so (n-1)! * 2 (the pair can swap). Three together: 3! inside the block.
- 2. NOT TOGETHER / ENDS
- Not together = n! - (together). A particular person at an end: 2 * (n-1)! for a line. Read 'at least one' as a split of cases if needed.
Overview
Glue, then subtract:
Together
n people, 2 must be together: treat the pair as one, so (n-1)! * 2 (the pair can swap). Three together: 3! inside the block.
Examples 2! for a pair · (n-1)! blocks · Word: vowels as a block
Not Together / Ends
Not together = n! - (together). A particular person at an end: 2 * (n-1)! for a line. Read 'at least one' as a split of cases if needed.
Examples All minus glued · Ends: 2 choices · Show both pieces
WHAT — RESTRICTIONS IN ARRANGEMENTS
Typical restrictions: some people must stand together (tie them as one block, then arrange the block internally). Some must not be together (total minus together). First/last slot fixed. Vowels together in a word. Meaning: rebuild the problem so the product rule still applies. Always start from the unrestricted total if the paper is a 'not together' question.
RULE — 1. TOGETHER
n people, 2 must be together: treat the pair as one, so (n-1)! * 2 (the pair can swap). Three together: 3! inside the block.
METHOD — 2. NOT TOGETHER / ENDS
Not together = n! - (together). A particular person at an end: 2 * (n-1)! for a line. Read 'at least one' as a split of cases if needed.
WORKED EXAMPLE
Worked: 5 people in a line, A and B together. glue AB as one: 4! ways to arrange blocks AB or BA: 2 4!2 = 48
How to answer a restrictions in arrangements question
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Worked example
- 5 people in a line, A and B together.
- glue AB as one: 4! ways to arrange blocks
- AB or BA: *2
- 4!*2 = 48
In simple words
Together uses a block. Not together is usually total minus together, not a new guess.
Key points
- together: glue as one super person
- not together: all - together
- block + inside
- complement
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Restrictions in Arrangements
Sticky Notes · Quick Revision
Core Idea
- Typical restrictions: some people must stand together (tie them as one block, then arrange the block internally).
- Some must not be together (total minus together).
- First/last slot fixed.
1. TOGETHER
- n people, 2 must be together: treat the pair as one, so (n-1)!
- 2 (the pair can swap).
- Three together: 3!
- inside the block.
2. NOT TOGETHER / ENDS
- Not together = n!
- (together).
- A particular person at an end: 2 * (n-1)!
- for a line.
Worked Example
- Worked: 5 people in a line, A and B together.
- glue AB as one: 4! ways to arrange blocks
- AB or BA: *2
- 4!*2 = 48
Common Misconception
- Together uses a block. Not together is usually total minus together, not a new guess.
FBISE Exam Check
- State the rule for restrictions in arrangements before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- Typical restrictions: some people must stand together (tie them as one block, then arrange the block internally).
- Some must not be together (total minus together).
- First/last slot fixed.
- Together uses a block. Not together is usually total minus together, not a new guess.
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