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Permutations & Combinations

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Permutations & Combinations

Quantitative Aptitude - Permutations & Combinations

Learn how to count arrangements (permutations) and selections (combinations) from a given set of items.

1. Basic Concepts

  • Permutation (Order matters): Ways of arranging r items from n distinct items.
  • Combination (Order doesn't matter): Ways of selecting r items from n items without arrangement.

2. Key Formulas

  • Permutations: \( P(n, r) = \frac{n!}{(n - r)!} \)
  • Combinations: \( C(n, r) = \frac{n!}{r!(n - r)!} \)
  • For identical objects: Total permutations = \( \frac{n!}{p_1! p_2! \dots p_k!} \)

3. Applications

  • Passwords, seating arrangements, committee formations, team selections.
  • Use permutations when arrangement matters (e.g., rankings).
  • Use combinations when only selection is required (e.g., team from group).

4. Examples

  • Number of 3-digit codes from digits 1-5 without repetition: \( P(5, 3) = 60 \)
  • Ways to choose 3 out of 6 people: \( C(6, 3) = 20 \)

5. Tips & Tricks

  • Read if order matters. If yes → permutation; if not → combination.
  • Factorial simplifications reduce calculation time.
  • Draw a tree diagram for visual help in selection/arrangement problems.

6. Mistakes to Avoid

  • Using permutation when question asks for selection only.
  • Ignoring constraints like repetition not allowed.

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