Solve algebraic permutation or combination counts
Translate an algebraic count into factorial notation, form an equation and keep only valid non-negative integer values.
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
If nC2 = 36, which equation should you solve?
What you will learn
- Simplify a permutation or combination formula into an equation
- Solve for the number of available items
- Reject negative or non-integer values that cannot describe a count
Translate the count before solving
If nC2 = 45, then n(n − 1)/2 = 45. Multiply by 2 and solve the quadratic, then check that a solution is an integer with n ≥ 2.
If nP2 = 42, then n(n − 1) = 42. Expand to a quadratic and retain the positive integer value that can represent the number of distinct objects.
Check the candidate in the original count
A quadratic equation may produce a negative root as well as the valid positive one. The number of available objects and the number selected must be non-negative integers, with r ≤ n.
Substitute the accepted value back into nCr or nPr. This catches arithmetic errors and confirms the count requested by the context.
Pause and explain
Why must a candidate solution for n be checked in a counting problem?
Worked example
A committee of 2 can be chosen in 45 ways from n people. Find n.
Show the worked solution
- Write nC2 = 45, so n(n − 1)/2 = 45.
- Rearrange: n² − n − 90 = 0 = (n − 10)(n + 9). Thus n = 10 or n = −9.
- A number of people cannot be negative, so retain n = 10; 10C2 = 45 confirms it.
Answer There are 10 people.
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.
An ordered pair from n distinct objects can be formed in 42 ways. Find n.
Give me a hint
Use nP2 = n(n − 1) and solve the resulting quadratic.
Compare my reasoning
- Set n(n − 1) = 42, so n² − n − 42 = 0.
- Factor: (n − 7)(n + 6) = 0, giving n = 7 or n = −6.
- Reject −6 and check 7P2 = 7 × 6 = 42.
n = 7 objects.
Look for these in your work
- Permutation formula used
- Both algebraic roots considered
- Valid count checked
A group of 2 can be selected in 66 ways. Find the number of available people.
Give me a hint
Use nC2 and retain only a whole-number set size.
Compare my reasoning
- Set n(n − 1)/2 = 66, so n² − n − 132 = 0.
- Factor: (n − 12)(n + 11) = 0.
- Reject n = −11; confirm 12C2 = 66.
12 people.
Look for these in your work
- Combination formula translated
- Invalid root rejected
- Original count verified
A school can make 120 different ordered arrangements of 3 distinct books selected from n books. Find n.
Give me a hint
Set nP3 = n(n − 1)(n − 2), then check the positive integer candidate.
Compare my reasoning
- Solve n(n − 1)(n − 2) = 120 for an integer n ≥ 3.
- Testing n = 6 gives 6 × 5 × 4 = 120.
- The product increases with integer n ≥ 3, so no other n satisfies it.
n = 6 books.
Look for these in your work
- Ordered arrangement formula written
- Valid integer candidate found
- Uniqueness checked in the allowed range
Common mistakes
- Keeping a negative root as a possible group size
- Solving the expanded equation but not checking the original count
- Forgetting the condition r ≤ n
- Treating a non-integer algebraic root as a possible number of objects
What can you explain now?
Can n = −4 be a solution for the number of items in nC2 = 10?
Compare with the explanation
No
A negative number cannot describe the size of the set, even if it appears as an algebraic root.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
After solving an algebraic counting equation, which domain and substitution checks establish that the answer is valid?
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.