Permutations & Combinations
Arrangements, selections, binomial theorem, and Pascal's triangle
Key Concepts
- Permutations count ordered arrangements; combinations count unordered selections
- Circular arrangements fix one position to remove rotational symmetry
- Stars and bars counts non-negative integer solutions
- Binomial coefficients satisfy Pascal's rule
Important Formulae
| Permutations / combinations | ⁿPᵣ = n!/(n−r)!; ⁿCᵣ = n!/(r!(n−r)!) |
| Circular arrangements | (n − 1)! for n distinct objects |
| Stars and bars | x₁+…+xₖ = n ⇒ C(n+k−1, k−1) solutions |
Quick Tips
- ⁿCᵣ = ⁿC₍ₙ₋ᵣ₎ — choose the smaller of r and n−r to compute faster.
- Decide first whether order matters: that picks permutation vs combination.
Sample Practice Questions
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Pascal's identity: C(n, r) = C(n-1, r-1) + ?
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Answer: C(n-1, r)
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The number of ways to arrange n distinct items is:
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Answer: n!
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How many 5-letter arrangements can be made from ABCDE, with A always first?
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Answer: 24
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C(n,0) + C(n,1) + ... + C(n,n) = ?
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Answer: 2^n
Practice Questions
Practise PRMO questions on Permutations & Combinations. Answers are revealed after each question.
Start Practice →Frequently Asked Questions
What are the key concepts in Permutations & Combinations for PRMO?
Permutations count ordered arrangements; combinations count unordered selections; Circular arrangements fix one position to remove rotational symmetry; Stars and bars counts non-negative integer solutions; Binomial coefficients satisfy Pascal's rule.
Which formulae are important for Permutations & Combinations in PRMO?
Permutations / combinations: ⁿPᵣ = n!/(n−r)!; ⁿCᵣ = n!/(r!(n−r)!); Circular arrangements: (n − 1)! for n distinct objects; Stars and bars: x₁+…+xₖ = n ⇒ C(n+k−1, k−1) solutions.
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