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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.
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Restrictions in Arrangements
What is it?
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.
In simple words
- 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.
Important terms
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.
What you should learn
- 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.
Examples
- 2! for a pair
- (n-1)! blocks
- Word: vowels as a block
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.
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.
Recommended for Class 11
Full school-level explanation
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.
Glue, then subtract:
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
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.
Remember
- block + inside
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.
Remember
- complement
Examples
- All minus glued
- Ends: 2 choices
- Show both pieces
How to answer a restrictions in arrangements question
How it works
- 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.
Remember
- together: glue as one super person
- not together: all - together
- 48
- complement
- block + inside
- complement
- together: glue as one super person
- not together: all - together
How the ideas connect
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
Deeper understanding
Go beyond memorising Restrictions in Arrangements. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
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 mistakes
Exam focus
Core Idea
1. TOGETHER
2. NOT TOGETHER / ENDS
Worked Example
Exam focus
Quick revision
- together: glue as one super person
- not together: all - together
- 48
- complement
- RESTRICTIONS IN ARRANGEMENTS: Typical restrictions: some people must stand together (tie them as one block, then arrange the block
- 1. TOGETHER: n people, 2 must be together: treat the pair as one, so (n-1)! * 2 (the pair can swap). Three togeth
Together uses a block. Not together is usually total minus together, not a new guess.
For long answers on Restrictions in Arrangements: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- together: glue as one super person
- not together: all - together
- 48
- complement
Important formulas
- together: glue as one super person
- not together: all - together
- 48
Common exam traps
- Together uses a block. Not together is usually total minus together, not a new guess.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Together uses a block. Not together is usually total minus together, not a new guess.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
Together uses a block. Not together is usually total minus together, not a new guess.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
together: glue as one super person
not together: all - together
48
complement
Typical restrictions: some people must stand together (tie them as one block, then arrange the block internally). Some must not be together (total minus togethe
- 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 togethe
- Example
- 2! for a pair
- Critical warning
- Together uses a block. Not together is usually total minus together, not a new guess.
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Exam Practice
Define
Q1. Define Restrictions in Arrangements.
Explain
Q2. Explain: TOGETHER.
Example
Q3. Give one clear example of Restrictions in Arrangements and say why it fits.
Mistake
Q4. State one common mistake students make about Restrictions in Arrangements.
Exam note
Q5. Write a short exam note on Restrictions in Arrangements: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.