JEE Main Maths · P&C PYQ
JEE Main Permutations & Combinations PYQ (2002–2025)
Permutations & Combinations (P&C) is a P2 Algebra chapter — ~4% JEE Main weightage with 1 question per session. Prerequisite for Probability (counting sample space) and Binomial Theorem. Problem-solving chapter with many word-problem variants.
Permutations & Combinations PYQs from 2002 to 2025 covering fundamental counting, permutations, combinations, circular permutations and distribution problems. Every solution identifies the correct counting technique.
Permutations & Combinations at a Glance
Key Sub-Topics & What's Tested
Fundamental Counting Principle
Multiplication principle: if A can happen in m ways and B in n ways, total = m × n. Addition principle for mutually exclusive cases.
Permutations (Arrangements)
P(n, r) = n!/(n-r)! for distinct arrangement. Permutations with repetition: n^r. Permutations of n objects with k identical: n!/(k₁! k₂! ...).
Combinations (Selections)
C(n, r) = n!/(r!(n-r)!). Combinations vs permutations distinction (order matters or not). Properties: C(n,r) = C(n,n-r).
Circular Permutations
n distinct objects around circular table = (n-1)!. Clockwise and anticlockwise count as different (unless bracelet/necklace — then divide by 2).
Permutations of Objects Not All Distinct
n objects with k₁ of type 1, k₂ of type 2, etc.: total = n!/(k₁! · k₂! · ...). Used in words with repeated letters.
Distribution Problems
Distributing identical objects into distinct boxes (stars and bars: C(n+r-1, r-1)). Distinct objects into distinct boxes.
Formation of Teams / Committees
Selecting r from n with constraints (must include specific person, must exclude, etc.). Use complement and case analysis.
Rank of a Word / Number
Alphabetical order of word arrangements, rank of given arrangement in dictionary order. Standard PYQ template.
Question Type Distribution
| Question Type | Share (approx) | Example Pattern |
|---|---|---|
| Word Arrangement / Rearrangement | 25% | How many arrangements of letters of "STATISTICS" start with S and end with T? |
| Committee / Team Formation | 20% | From 6 men and 4 women, how many committees of 5 contain at least 2 women? |
| Circular Arrangements | 15% | In how many ways can 6 people be seated around a round table if 2 must sit together? |
| Distribution Problems | 15% | How many ways to distribute 10 identical balls among 3 boxes if each box gets at least 1? |
| Seat/Route Arrangements | 15% | Number of ways to arrange 5 boys and 5 girls alternately in a row. |
| Rank in Dictionary Order | 10% | Find rank of "JEE" when letters are arranged in alphabetical order. |
How to Solve Permutations & Combinations PYQs
- 1Identify if order matters. "Arrangement" or "ordering" = permutation. "Selection" or "group" = combination. First step in every P&C problem.
- 2Use multiplication principle for sequential choices. If problem has multiple stages, multiply counts for each stage. Break complex problems into clear stages.
- 3For "at least"/"at most" conditions, use complement. At least 1 = Total - None. At most 3 = None + 1 + 2 + 3 individually summed.
- 4Circular arrangements: fix one object. Breaking rotational symmetry by fixing one person = (n-1)! for n distinct people. Avoids over-counting.
- 5For distribution, use stars-and-bars. n identical objects into r distinct boxes (each can be empty): C(n+r-1, r-1). Each box ≥ 1: C(n-1, r-1).
Common Mistakes That Cost Marks
- Permutation vs combination confusion. C(5,3) = 10 (selections). P(5,3) = 60 (arrangements = 6× selections). Order matters → permutation.
- Overcounting when cases overlap. Union of counts must subtract overlap: |A ∪ B| = |A| + |B| - |A ∩ B|. Common mistake in "contains specific person" problems.
- Wrong formula for circular permutations. n distinct people around table = (n-1)!, not n!. Fixing one person removes rotational duplicates.
- Missing repetition factor. Arrangements of "AABBC" = 5!/(2!·2!) = 30, not 5! = 120. Divide by factorial of each repeated letter count.
- Forgetting constraints in committee problems. "At least 2 women" = C(women, 2)·C(rest) + C(women, 3)·C(rest) + ... Don't just use C(n,r).
Related JEE Main Practice
Frequently asked questions
How many P&C PYQs should I solve?
Target 50–70 PYQs across 2010–2025. Given the problem-pattern density, focused practice builds strong case-analysis intuition.
Is P&C harder than Sequences & Series?
Conceptually similar — both are Algebra problem-solving chapters. P&C has more word-problem variants. Sequences is more formula-bound. Master both together.
Do I need to memorise circular permutation formulas?
Yes — (n-1)! for n distinct around table. (n-1)!/2 for necklace/bracelet. For n objects with k adjacent together, use (n-k+1)!·k!. Standard patterns.
What's the most-tested P&C pattern?
Word arrangement problems with specific letter conditions (starts with, contains all vowels together, etc.). Appears in 3-4 PYQs per 10-year window.
How does P&C connect with Probability?
Heavily — probability often requires counting the sample space (total outcomes) and favourable outcomes. Strong P&C foundation directly accelerates Probability problem-solving.
Is Pigeonhole Principle tested?
Rarely in JEE Main — maybe once per 10-year window. Conceptual "is it possible that..." type questions. Not a high-priority topic.
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