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PERMUTATIONS nPr
A permutation is an arrangement (order matters). The number of ways to arrange r objects taken from n different objects is nPr = n! / (n-r)! = n(n-1)...(n-r+1). nPn = n!. Meaning: first slot n choices, second n-1, ... r slots. If order does not matter, that is a combination (later).
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PERMUTATIONS nPr
What is it?
A permutation is an arrangement (order matters). The number of ways to arrange r objects taken from n different objects is nPr = n! / (n-r)! = n(n-1)...(n-r+1). nPn = n!. Meaning: first slot n choices, second n-1, ... r slots. If order does not matter, that is a combination (later).
In simple words
- A permutation is an arrangement (order matters).
- The number of ways to arrange r objects taken from n different objects is nPr = n!
- / (n-r)!
- nPr needs r <= n. 5P7 is not a school permutation of distinct objects.
Important terms
PERMUTATIONS nPr: A permutation is an arrangement (order matters). The number of ways to arrange r objects taken from n different objects is nPr = n! / (n-r)! = n(n-1)...(n-r+1). nPn = n!. Meaning: first slot n choices, second n-1, ... r slots. If order does not matter, that is a combination (later).
1. FORMULA: 5P2 = 5*4 = 20. 5P5=120. 5P0=1 (empty arrangement). Need r <= n. Proof: n choices, then n-1, ... r factors.
2. WORDING: 'Arrange', 'line up', 'first, second, third', 'codes', 'passwords order' -> permutation. President, VP, secretary from 10 people: 10P3.
What you should learn
- State the main ideas and notation used in permutations npr.
- Apply the rules of permutations npr in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Examples
- nP1 = n
- nPn = n!
- nP0 = 1
FORMULA
5P2 = 5*4 = 20. 5P5=120. 5P0=1 (empty arrangement). Need r <= n. Proof: n choices, then n-1, ... r factors.
WORDING
'Arrange', 'line up', 'first, second, third', 'codes', 'passwords order' -> permutation. President, VP, secretary from 10 people: 10P3.
Recommended for Class 11
Full school-level explanation
A permutation is an arrangement (order matters). The number of ways to arrange r objects taken from n different objects is nPr = n! / (n-r)! = n(n-1)...(n-r+1). nPn = n!. Meaning: first slot n choices, second n-1, ... r slots. If order does not matter, that is a combination (later).
Formula, then a lineup:
WHAT — PERMUTATIONS nPr
A permutation is an arrangement (order matters). The number of ways to arrange r objects taken from n different objects is nPr = n! / (n-r)! = n(n-1)...(n-r+1). nPn = n!. Meaning: first slot n choices, second n-1, ... r slots. If order does not matter, that is a combination (later).
RULE — 1. FORMULA
5P2 = 5*4 = 20. 5P5=120. 5P0=1 (empty arrangement). Need r <= n. Proof: n choices, then n-1, ... r factors.
METHOD — 2. WORDING
'Arrange', 'line up', 'first, second, third', 'codes', 'passwords order' -> permutation. President, VP, secretary from 10 people: 10P3.
WORKED EXAMPLE
Worked: 7P3. 7P3 = 7*6*5 = 210 or 7!/(4!) = 504024 = 210
FORMULA
5P2 = 5*4 = 20. 5P5=120. 5P0=1 (empty arrangement). Need r <= n. Proof: n choices, then n-1, ... r factors.
Remember
- r factors falling
Examples
- nP1 = n
- nPn = n!
- nP0 = 1
WORDING
'Arrange', 'line up', 'first, second, third', 'codes', 'passwords order' -> permutation. President, VP, secretary from 10 people: 10P3.
Remember
- order words
Examples
- Gold silver bronze
- Different positions
- Not a committee yet
How to answer a permutations npr 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
- order matters
- nPr = n! / (n-r)!
- 210
- r <= n
- r factors falling
- order words
- order matters
- nPr = n! / (n-r)!
How the ideas connect
Worked: 7P3. 7P3 = 7*6*5 = 210 or 7!/(4!) = 504024 = 210
Deeper understanding
Go beyond memorising PERMUTATIONS nPr. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: 7P3. 7P3 = 7*6*5 = 210 or 7!/(4!) = 504024 = 210
Common mistakes
Exam focus
Core Idea
1. FORMULA
2. WORDING
Worked Example
Exam focus
Quick revision
- order matters
- nPr = n! / (n-r)!
- 210
- r <= n
- PERMUTATIONS nPr: A permutation is an arrangement (order matters). The number of ways to arrange r objects taken from
- 1. FORMULA: 5P2 = 5*4 = 20. 5P5=120. 5P0=1 (empty arrangement). Need r <= n. Proof: n choices, then n-1, ... r f
nPr needs r <= n. 5P7 is not a school permutation of distinct objects.
For long answers on PERMUTATIONS nPr: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- order matters
- nPr = n! / (n-r)!
- 210
- r <= n
Important formulas
- order matters
- nPr = n! / (n-r)!
- 210
Common exam traps
- nPr needs r <= n. 5P7 is not a school permutation of distinct objects.
- 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
nPr needs r <= n. 5P7 is not a school permutation of distinct objects.
- 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
nPr needs r <= n. 5P7 is not a school permutation of distinct objects.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
order matters
nPr = n! / (n-r)!
210
r <= n
A permutation is an arrangement (order matters). The number of ways to arrange r objects taken from n different objects is nPr = n! / (n-r)! = n(n-1)...(n-r+1).
nPr = n! / (n-r)!
- Definition
- A permutation is an arrangement (order matters). The number of ways to arrange r objects taken from n different objects is nPr = n! / (n-r)! = n(n-1)...(n-r+1).
- Formula
- nPr = n! / (n-r)!
- Example
- nP1 = n
- Critical warning
- nPr needs r <= n. 5P7 is not a school permutation of distinct objects.
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Exam Practice
Define
Q1. Define PERMUTATIONS nPr.
Explain
Q2. Explain: FORMULA.
Example
Q3. Give one clear example of PERMUTATIONS nPr and say why it fits.
Mistake
Q4. State one common mistake students make about PERMUTATIONS nPr.
Exam note
Q5. Write a short exam note on PERMUTATIONS nPr: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.