Count ordered arrangements without repetition
Use nPr to count ordered arrangements of r distinct items selected from n when no item can be repeated.
Section 11 · Permutations and combinations
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What does nPr count?
What you will learn
- Use nPr = n!/(n − r)! for ordered arrangements
- Count choices position by position without replacement
- Check that r is a non-negative integer no greater than n
Each position is a different role
For an ordered arrangement of r distinct items from n, the first position has n choices, the second n − 1, and so on until r positions are filled. Therefore nPr = n(n − 1)…(n − r + 1) = n!/(n − r)!.
The notation nPr is read as permutations of n items taken r at a time. Here 0 ≤ r ≤ n and items are not repeated; if r = 0, the expression gives one empty arrangement because 0! = 1.
Count distinct ordered outcomes only
Arrange four of seven books on a shelf: 7P4 = 7 × 6 × 5 × 4. Each position is visible and no book is used twice.
Do not use nPr for a group with no order, a circular arrangement, or a sequence that permits repeated items. Those are different counting models and repetition or circular arrangements are outside this 0606 topic.
Pause and explain
Which expression counts 3 ordered positions from 8 distinct people?
Worked example
How many ways can 4 different books be arranged on a shelf if they are selected from 7 different books?
Show the worked solution
- The shelf positions are ordered, so use a permutation.
- The four positions have 7, 6, 5 and 4 choices, with no repeats.
- Multiply: 7P4 = 7 × 6 × 5 × 4 = 840.
Answer 840 arrangements.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
How many ways can 4 of 7 different books be placed in four labelled display positions?
Give me a hint
A labelled position makes each placement role distinct.
Compare my reasoning
- Use 7P4.
- Expand as 7 × 6 × 5 × 4.
- Multiply to get 840.
840 placements.
Look for these in your work
- Order recognised
- No repeated book
- Ordered product correct
A race has 9 finalists. How many different results can fill first through fourth place?
Give me a hint
There are four ordered places, with no runner appearing twice.
Compare my reasoning
- The choices are 9 for first, 8 for second, 7 for third and 6 for fourth.
- Write this as 9P4.
- Multiply 9 × 8 × 7 × 6.
9P4 = 3024 results.
Look for these in your work
- Four distinct positions counted
- Choices decrease by one
- Product evaluated
A five-symbol access sequence uses distinct symbols chosen from digits 1 to 7. How many sequences are possible?
Give me a hint
The five slots are ordered; no digit may appear twice.
Compare my reasoning
- There are 7 choices for the first slot.
- There are 6, 5, 4 and 3 choices for the remaining slots.
- Multiply: 7P5 = 7 × 6 × 5 × 4 × 3.
2520 sequences.
Look for these in your work
- Order matters
- No repetition applied
- All five positions counted
Common mistakes
- Using nCr when positions or roles are distinct
- Using n! even though only r of the n items are arranged
- Allowing an item to be reused when the question says distinct
- Using nPr when r > n
What can you explain now?
How many ordered pairs can be made from 6 distinct objects without repetition?
Compare with the explanation
6P2 = 30
There are 6 choices for the first position and 5 for the second.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why does nPr multiply a decreasing sequence of choices, and how does that differ from arranging all n items?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.