MathematicsChapter 6: Permutation and Combination
Not Together / Ends
Not together = n! - (together).
Learning objectives
- State the rule or meaning of not together / ends in easy words.
- Apply not together / ends in a clear step-by-step solution.
- Check the conditions, working, and final answer for accuracy.
Definition
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.
Formula
together = n
Where / symbols
- 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.
- NOT TOGETHER / ENDS
- Not together = n!
Overview
Understand the idea, then apply it:
Main Idea
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
Exam Use
Write the rule first: Not together = n! Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Examples State the central idea · Use one accurate detail · Check the final response
WHAT — 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.
WHY — Why it matters
This rule is a separate learning unit inside restrictions in arrangements. It can be defined, applied, and checked in its own FBISE question.
HOW — How to answer
Write the rule first: Not together = n! Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
How to answer a not together / ends question
- Read the exact command word and identify what the question asks.
- State the matching idea, rule, or meaning before adding detail.
- Use the unit evidence or mathematical working that directly supports the answer.
- Check accuracy, relevance, and answer length before finishing.
In simple words
Together uses a block. Not together is usually total minus together, not a new guess.
Key points
- complement
- together: glue as one super person
- not together: all - together
- focused answer
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Not Together / Ends
Sticky Notes · Quick Revision
Core Idea
- Not together = n!
What to Know
- 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.
Example or Evidence
- All minus glued
- Ends: 2 choices
- Show both pieces
How to Answer
- Write the rule first: Not together = n! Substitute carefully, show one step per line, and check signs, restrictions, units, and the final result.
Common Error
- Together uses a block. Not together is usually total minus together, not a new guess.
FBISE Exam Check
- State the rule, show the working, and check the answer.
60-second revision
- Not together = n!
- All minus glued
- Ends: 2 choices
- State the rule, show the working, and check the answer.
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