How to Solve Permutation and Combination Problems
Solve permutation and combination aptitude problems — order-matters test, formulas, multiplication principle — with a worked example and practice questions.
Expected Interview Answer
Permutations count arrangements where order matters (nPr = n!/(n−r)!), while combinations count selections where order does not matter (nCr = n!/(r!(n−r)!)); the deciding question is always "does swapping the order create a new outcome?"
If arranging 3 people in a race (1st, 2nd, 3rd) counts different orders as different outcomes, that is a permutation. If choosing any 3 people for a committee treats all orderings as the same group, that is a combination — hence nCr = nPr / r!, dividing out the r! ways to order each group. The multiplication principle underlies both: if a task splits into independent stages, multiply the counts of ways to do each. Watch for identical items (divide by their factorial) and circular arrangements (divide by n, or 2n for reflections).
- One question — "does order matter?" — picks the right formula
- The multiplication principle unifies multi-stage counting
- Correction factors (identical items, circular arrangements) prevent overcounting
AI Mentor Explanation
Choosing 11 players from a 15-man squad for a match is a combination — nC11 — because the *team* is the same regardless of who’s listed first. Deciding the actual batting order for those 11 is a permutation — 11! — because swapping two batters’ positions creates a genuinely different lineup. Selection ignores order (combination); arranging cares about order (permutation) — that single distinction is the whole topic.
Worked example
Combination (committee)
- 8C3 = 8!/(3!5!) = 56
Permutation (3 roles)
- 8P3 = 8!/5! = 336
Relation
- nPr = nCr × r!
- 336 = 56 × 6
Step-by-Step Explanation
Step 1
Ask: does order matter?
Order matters → permutation; order irrelevant → combination.
Step 2
Apply the formula
nPr = n!/(n−r)!; nCr = n!/(r!(n−r)!) = nPr / r!.
Step 3
Use the multiplication principle
Independent stages multiply: total ways = product of each stage’s count.
Step 4
Correct for special cases
Divide by k! for k identical items; divide by n (or 2n) for circular arrangements.
What Interviewer Expects
- Correct order-matters test to pick permutation vs combination
- Accurate use of factorial formulas
- Application of the multiplication principle for multi-stage problems
- Handling identical items and circular arrangements correctly
Common Mistakes
- Using nPr when the problem actually asks for a selection (combination)
- Forgetting to divide by r! when converting arrangements to selections
- Not dividing by k! when items in the group are identical
- Mishandling circular arrangements by not dividing by n
Best Answer (HR Friendly)
“The whole topic comes down to one question: does the order matter? If yes, it’s a permutation — n!/(n−r)!. If no, it’s a combination — n!/(r!(n−r)!), which is just the permutation divided by r! to remove the duplicate orderings of each group. For multi-stage problems, multiply the counts of independent choices together.”
Follow-up Questions
- How do you count arrangements with some identical items?
- How does a circular arrangement formula differ from a linear one?
- How do you count combinations with repetition allowed?
- How would you count the number of ways to distribute distinct objects into groups?
MCQ Practice
1. How many ways can 5 distinct books be arranged on a shelf?
5! = 120, since every book position matters (permutation).
2. How many ways can a committee of 3 be chosen from 6 people?
6C3 = 6!/(3!3!) = 20; order within the committee does not matter.
3. nPr and nCr are related by which formula?
nPr counts ordered selections; dividing out the r! orderings of each group gives nCr, so nPr = nCr × r!.
Flash Cards
Permutation formula? — nPr = n! / (n−r)! — order matters.
Combination formula? — nCr = n! / (r!(n−r)!) — order does not matter.
Relation between them? — nPr = nCr × r!.
Circular arrangement of n items? — (n−1)! ways, since rotations are equivalent.