Permutations and Combinations

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Permutations and Combinations — Key Facts for JEE Main Fundamental principle of counting: if task A can be done in m ways and task B in n ways, then A followed by B can be done in m × n ways Addition principle: if tasks are exclusive (either A or B), total = m + n Permutation (arrangement): nP_r = n!/(n−r)! — order matters Combination (selection): nC_r = n!/[r!(n−r)!] — order does not matter Key: nC_r = nC_{n−r} (complementary property) ⚡ Exam tip: JEE Main P&C problems are tricky — always ask: “Does order matter?” If yes → permutations; if no → combinations!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Permutations and Combinations — JEE Main Study Guide

Counting principles:

• Arrangement of n distinct objects: n! ways
• Circular arrangement: (n−1)! for distinct objects around a circle (clockwise vs anticlockwise considered same)
• Arrangement in a line with some identical objects: n!/[n₁!n₂!…]

Special arrangements:

• Number of ways to arrange n letters of word “BANANA”: B(1), A(3), N(2) → 5!/(1!3!2!) = 20
• Rank of a word in dictionary: arrange letters alphabetically, count words before the target

Restricted permutations:

• Objects in a row, A must be at one end: 2 × (n−1)! (A at left OR right end)
• Objects in a row, A and B must be together: treat AB as one unit → 2 × (n−1)! (A left of B OR B left of A)
• Objects in a row, A and B must NOT be together: total − together = n! − 2×(n−1)! = (n−2)(n−1)!

Derangements: Number of ways n objects can be arranged so that no object is in its original position: D_n = n![1 − 1/1! + 1/2! − 1/3! + … + (−1)^n/ n!]; also D_n = (n−1)(D_{n−1} + D_{n−2}) with D₁ = 0, D₂ = 1 Alternatively: D_n = round(n!/e)

Partition problems:

• Divide n identical objects into r groups (allow empty groups): C(n+r−1, r−1)
• Divide n distinct objects into r groups (allow empty): r^n (each object has r choices)
• Divide n distinct objects into r nonempty groups: use Stirling numbers of second kind S(n, r) or inclusion-exclusion

Distribution problems:

• Distribute n distinct objects to r distinct boxes (boxes can be empty): r^n

• Distribute n distinct objects to r distinct boxes (no box empty): use inclusion-exclusion: r^n − C(r,1)(r−1)^n + C(r,2)(r−2)^n − …

• Distribute n identical objects to r distinct boxes: C(n+r−1, r−1)

• Key formula: nC_r = n!/[r!(n−r)!]; nP_r = n!/(n−r)!

• Common trap: For “at least one” problems, use total − none = total − (all fail) — inclusion-exclusion often needed

• Exam weight: 1–2 questions per year (4–8 marks); high difficulty, high scoring for those who understand the logic

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Permutations and Combinations — Comprehensive JEE Main Notes

Inclusion-Exclusion Principle: | A ∪ B | = | A | + | B | − | A ∩ B | | A ∪ B ∪ C | = | A | + | B | + | C | − | A∩B | − | B∩C | − | C∩A | + | A∩B∩C |

Applications of inclusion-exclusion:

• Count numbers from 1 to 1000 not divisible by 2, 3, or 5: total 1000 − floor(1000/2) − floor(1000/3) − floor(1000/5) + floor(1000/6) + floor(1000/10) + floor(1000/15) − floor(1000/30)
• “At least one” complement: total − none

The “stars and bars” method: For distributing n identical items into r distinct boxes:

• Allow empty boxes: C(n + r − 1, r − 1)
• No box empty: C(n − 1, r − 1) (first give 1 to each box, remaining n−r identical items → C((n−r)+r−1, r−1) = C(n−1, r−1))

Grouping problems:

• Divide n objects into groups of sizes a, b, c (order of groups doesn’t matter): n!/[a!b!c!] when a+b+c=n and groups are distinct
• If groups are also of different sizes: the multiplier depends on whether groups have same size
• If groups are identical (just “divide into 3 groups of sizes a, b, c where a, b, c are equal): then divide by factorial of equal group sizes

Arrangement with repetitions: Number of arrangements of n items with p identical of type 1, q identical of type 2, r identical of type 3…: n!/[p!q!r!…]

Selection with conditions:

• Select r objects from n, where a particular object is always included: C(n−1, r−1)
• Select r objects from n, where a particular object is never included: C(n−1, r)
• Select r objects from n, where two particular objects are either both included or both excluded: C(n−2, r−2) + C(n−2, r)

Multinomial theorem connection: Number of ways to distribute n distinct objects into r distinct groups of sizes n₁, n₂, …, n_r (where n₁+n₂+…+n_r = n) is: n!/(n₁! n₂! … n_r!) This is also the coefficient of a^{n₁} b^{n₂} … in (a + b + …)^n

Key combinatorial identities:

• Pascal’s identity: C(n, r) = C(n−1, r) + C(n−1, r−1)
• Vandermonde’s identity: Σ C(m, k) C(n, r−k) = C(m+n, r)
• C(n, k) = n/k · C(n−1, k−1)
• C(n, 0) + C(n, 1) + … + C(n, n) = 2^n

Circular permutations:

• n distinct people around a round table: (n−1)! arrangements
• If clockwise/anticlockwise are different: 2(n−1)!
• If rotations are considered same: (n−1)!/2 for necklaces (but usually not needed in JEE)
• n distinct beads in a necklace: (n−1)!/2

Bridge crossing and similar: Think carefully about whether arrangements are being counted once or twice Example: “How many ways can 5 men and 5 women be seated at a round table so that no two women are adjacent?” Treat men first: fix one man → arrange 4 others in (4−1)! = 6 ways → 5 men occupy 5 alternate seats → 5 women choose 5 remaining seats: 5! ways → total = 6 × 120 = 720

• Remember: Order matters → permutation (nP_r); order doesn’t matter → combination (nC_r); to handle “at least one” or “none”, use complement: total − cases where condition fails; for distributing identical items, use stars and bars: C(n+r−1, r−1)
• Previous years: “Number of ways to select 3 consonants and 2 vowels from 7 consonants and 4 vowels” [2023]; “Number of ways to form a committee with at least 3 men from 5 men and 4 women” [2024]; “Arrange word ASSASSINATION” [2024]

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📊 JEE Main Exam Essentials

Detail	Value
Questions	90 (30 per subject)
Time	3 hours
Marks	300 (90 per subject)
Section	Physics (30), Chemistry (30), Mathematics (30)
Negative	−1 for wrong answer
Mode	Computer-based

🎯 High-Yield Topics for JEE Main Mathematics

• Calculus (Differentiation + Integration) — ~35 marks combined
• Coordinate Geometry (straight lines, circles, conics) — ~20 marks
• Algebra (Complex Numbers, Quadratics, P&C, Probability) — ~25 marks
• Trigonometry + Inverse Trigonometry — ~15 marks
• Vector + 3D — ~15 marks

📝 Previous Year Question Patterns

• P&C: 1–2 questions per year, 4–8 marks
• Common patterns: arrangements with restrictions, distribution of identical/distinct objects, committee formation, inclusion-exclusion
• Weight: high difficulty, high rewards — master it!

💡 Pro Tips

• The most common error is confusing permutations and combinations — ask “does order matter here?”
• For “at least one” problems, always try complement: total minus none
• Stars and bars: n identical objects into r distinct boxes (empty allowed) → C(n+r−1, r−1)
• For arrangements with restrictions like “A and B must sit together”, treat AB as one unit first
• When asked about seating around a round table, always divide by n for rotational symmetry
• For word arrangements, separate identical letters first, then arrange total letters dividing by factorials of each repeated letter
• Inclusion-exclusion is essential for “not divisible by” type questions
• Many JEE Main P&C problems can be solved by first principles — don’t jump to formulas

🔗 Official Resources

• NTA Official JEE Main
• JEE Main Syllabus PDF

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