Questions
4 questions per paper
Difficulty
Medium
Importance
Core topic for logical reasoning
Overview
Permutations and Combinations (P&C) form the backbone of counting techniques and probability theory. Mastering this topic is essential for competitive exams as it tests logical reasoning, systematic listing, and the ability to distinguish between ordered arrangements and unordered selections.
Fundamental Principle of Counting
This principle provides the foundation for all counting problems by defining the multiplication and addition rules. Understanding when to multiply independent events versus adding mutually exclusive options is crucial for solving multi-stage problems.
- Fundamental Principle of Multiplication: If an event can occur in m ways and another in n ways, their sequence occurs in m × n ways
- Fundamental Principle of Addition: If events are mutually exclusive, total ways = m + n
- Factorial notation definition: n! = n × (n-1) × (n-2) × ... × 1
- Definition of 0! = 1
- Constraint handling: Always address restricted positions or items first
Permutations (Arrangements)
Permutations deal with ordered arrangements where the sequence of elements matters significantly. In exam scenarios, look for keywords like 'arranging', 'seating', or 'forming numbers' to identify permutation problems.
- Permutation formula: P(n, r) = n! / (n-r)!
- Arrangement of n items: n!
- Circular permutation: (n-1)!
- Permutations with identical items: n! / (p!q!r!)
- Permutation of n items taken r at a time with repetition: n^r
Combinations (Selections)
Combinations involve selecting items where order is irrelevant, focusing solely on the composition of the group. This is the standard approach for committees, teams, or selecting subsets from a larger pool.
- Combination formula: C(n, r) = n! / [r!(n-r)!]
- Symmetry property: C(n, r) = C(n, n-r)
- Pascal's identity: C(n, r) + C(n, r-1) = C(n+1, r)
- Selection of at least one item: 2^n - 1
- Relation: P(n, r) = C(n, r) × r!
Formula Sheet
P(n, r) = n! / (n-r)!
C(n, r) = n! / (r!(n-r)!)
Circular arrangement = (n-1)!
C(n, r) = C(n, n-r)
Exam Tip
Always identify if the problem requires 'ordering' or 'selection' first; if the answer changes by swapping two items, use permutations, otherwise use combinations.
Common Mistakes
- Confusing Permutations with Combinations: Using the arrangement formula when only a selection is required
- Ignoring the 'at least' condition: Failing to calculate the total minus the 'none' case
- Overcounting in circular arrangements: Forgetting to divide by n when items can be rotated
More Revision Notes
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