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.
Topics
- Number System
- H.C.F. & L.C.M.
- Decimal Fractions
- Simplification
- Square Roots & Cube Roots
- Average
- Problems on Numbers
- Problems on Ages
- Surds & Indices
- Percentage
- Operations on Numbers
- Profit & Loss
- Ratio & Proportion
- Partnership
- Chain Rule
- Time & Work
- Pipes & Cistern
- Time & Distance
- Problems on Trains
- Boats & Streams
- Alligation or Mixture
- Simple Interest
- Compound Interest
- Logarithms
- Area
- Volume & Surface Areas
- Mensuration
- Races & Games of Skill
- Calendar
- Clocks
- Stocks & Shares
- Permutations & Combinations
- Probability
- True Discount
- Banker’s Discount
- Heights & Distances
- Odd Man Out & Series
- Algebra
- Divisibility Rules & Remainder Theorem
- Factorials & Power Cycles
- Progressions (A.P., G.P., H.P.)
- Work & Wages
- Set Theory / Venn Diagrams
- Geometry (Lines, Angles, Triangles, Polygons)
- Coordinate Geometry
- Data Interpretation
- Trigonometry
- Statistics (Mean, Median, Mode)
- Arithmetic Operations
- Quadratic Equations