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Permutations & Combinations

P&C full marks come from one habit: write a counting model sentence (order? replacement? constraints?) before any nPr/nCr — then finish in 60–90 seconds.

Portion AlgebraFocus High-accuracy Algebra bank when model sentence is lockedTarget 60–90sPortion share 37.2%Trend Falling share
Advanced P&C is modelling, not formula hunting. If order, replacement, and constraints are frozen in words, the arithmetic is usually short. This page is only P&C — not mixed with Sets/Relations/Sequences. P&C scoring is the model sentence; nPr/nCr only print the answer.

How Advanced asks this

  • Constrained selections (consecutive / distinct / at least one)
  • Arrangements with repetition or identical objects
  • Stars-and-bars / positive integer solutions
  • Counting that feeds a probability stem (same model first)
  • Matching: counting scenario ↔ expression

Traps that leak marks

  • Using nCr when seats/order matter
  • Stars-and-bars without positivity shift
  • Changing ‘order matters’ mid-solution
  • Double-counting after case splits that overlap
  • Stars-and-bars on distinct objects (or n^r on identical)
  • Overlapping cases that share some counted tuples
  • Forgetting division by symmetries for identical items

01. Why P&C stands alone: it joins Probability often, but the counting skill is separate. Mixing it with relations or AP/GP in one revision block creates weak classification under pressure.

02. Common process: classify the sample object → write model sentence → choose P/C/stars-and-bars/complement → compute → overcount check.

03. Quick formulas only work after the model is correct. Wrong model + right formula = wrong answer every time.

04. Full marks mindset: a correct model in 20s beats a fast nCr guess. Speed comes from classification language, not from memorising more identities.

05. Advanced lane: ‘at least / at most / none’ wording almost always wants complement or a clean case split — decide which in the first 20s from the model sentence.

06. Distribution vs arrangement: identical objects and distinct boxes change the formula family; say ‘identical/distinct’ out loud before stars-and-bars or n^r.

Official · 2026

Syllabus for this topic

Official syllabus points for Permutations & Combinations only — not the full Algebra portion map. Portion hub: Algebra · Algebra analysis.

Syllabus on this page is topic-scoped: Permutations & Combinations. The parent portion (Algebra) also covers other topics — use the portion hub for the full map.

Must cover (this topic)

  • Permutations and combinations

Gaps / watch-outs

No topic-specific exclusions flagged — finish the unit points below for P&C.

Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf

Unit 01

Permutations & Combinations

high

Exam focusConstrained counting models (order, replacement, identical objects) before nPr/nCr.

Syllabus points

  • Permutations and combinations

Full Mathematics syllabus →

Cross-topic stems

Mixed joins for this topic

Advanced rarely asks a chapter in isolation. These are tagged multi-idea Math stems that touch Permutations & Combinations — practise the knowledge bridge, not only the single-chapter drill.

No tagged mixed stem matched this topic’s keywords yet. Still revise the joins listed under Trends, and scan the mixed joins board for Calculus / Coordinate / Algebra bridges.

All mixed joins →

Why methods matter

Common methods that change your score

P&C is modelling English into maths. These methods make the model sentence non-negotiable — once that sticks, nPr/nCr become easy endings, not starting points.

01

Model sentence lock

This sentence is the method. Formulae are only the last line — believe that and P&C changes.

Do thisWrite order/replacement/constraint in one sentence before formulae.

Why it works

Kills the #1 error: using nCr when order matters (or the reverse).

See it once

‘Number of 3-digit PINs with distinct digits’ → order matters, no replacement → 10P3 (or 10×9×8). Sentence decides P vs C.

02

Gap / shift transform

Turns scary constraints into ordinary combinations — that feels like a superpower in the hall.

Do thisFor consecutive or positivity constraints, redefine variables first.

Why it works

Turns constrained counting into plain combinations.

See it once

No two consecutive from 1…20 choose 3: yi=xi−(i−1) → C(18,3). The transform is the knowledge.

03

Case split only when forced

Stops overlapping cases that look almost right but are wrong.

Do thisPartition into disjoint cases; add. Never overlap cases.

Why it works

Prevents partial overcount that looks ‘almost right’.

See it once

‘At least one girl in a committee’ → total − all boys (complement) often beats many girl-count cases.

04

Identical-object division

Prevents counting indistinguishable arrangements as distinct.

Do thisIf objects are identical in type, divide by factorials of repeats.

Why it works

Stops treating indistinguishable arrangements as distinct.

See it once

MISSISSIPPI letters: 11!/(4!4!2!). Write the multiset before computing.

Processed examples

Sample questions · full working

Read the impact line first. Then notice which family/model was chosen. The solution is written the way your rough sheet should look in the hall.

01

standardFamily · Stars-and-bars / integer solutions

Sample question

Number of positive integer solutions of x+y+z=10.

Positivity shift is the model — without it, stars-and-bars is applied wrong.

Model chosenyi=xi−1≥0 → C(9,2).

Processed solution

  1. xi≥1 ⇒ yi=xi−1 ≥ 0.
  2. y1+y2+y3=7.
  3. Non-neg solutions: C(7+3−1,3−1)=C(9,2)=36.

Answer36

Takeaway: State ≥0 vs ≥1 before the combination.

02

standardFamily · Gap / non-consecutive selection

Sample question

Number of ways to choose 3 integers from 1…20 with no two consecutive.

Gap transform turns a scary constraint into ordinary C(n,k).

Model chosenyi=xi−(i−1) → choose 3 from 18.

Processed solution

  1. Map x1<x2<x3 with gaps ≥2 to strictly increasing y_i.
  2. yi = xi − (i−1) ∈ {1,…,18}.
  3. Count = C(18,3)=816.

Answer816

Takeaway: Write the transform sentence — that is the knowledge.

03

twistFamily · Stars-and-bars / integer solutions

Sample question

Count positive integer solutions of x₁ + x₂ + x₃ + x₄ = 20 with x₁ odd, x₂ ≥ 2 and 1 ≤ x₃ ≤ 4.

Parity + lower/upper bounds must enter the model sentence before any C(n, k).

Model chosenEncode oddness and bounds, then sum over the bounded variable.

Processed solution

  1. Set x₁ = 2a + 1 (a ≥ 0), x₂ = b + 2 (b ≥ 0), x₄ = d + 1 (d ≥ 0), x₃ = c ∈ {1, 2, 3, 4}.
  2. Then 2a + b + d = 16 − c.
  3. For n = 16 − c, #solutions = Σ_{a=0}^{⌊n/2⌋}(n − 2a + 1).
  4. Counts for c = 1, 2, 3, 4: 72, 64, 56, 49.
  5. Total = 72 + 64 + 56 + 49 = 241.

Answer241

Takeaway: Translate positivity, parity, and bounds before placing stars/bars.

Reusable models

Common question families

Advanced stems cluster. Name the family in 10–15s, write the fast model, then run the same path. Do not invent a new plan each time.

01

Arrange vs choose

Looks likePINs, rankings, committees, teams

Fast modelOne sentence: order matters? → P or × sequential; else C.

Same solve path

  1. Model sentence (order / replacement / identical).
  2. Pick P, C, or multiset n!/(n₁!…).
  3. Compute.
  4. Overcount check.

Rough skeleton

  • Sentence
  • Tool
  • Number

Why faster

Classification in 15s prevents the classic P↔C swap.

Refuse this

Do not start with a formula before the sentence.

02

Stars-and-bars / integer solutions

Looks likex₁+…+x_k=n with xi≥0 or ≥1

Fast modelState ≥0 vs ≥1; if ≥1 shift y_i=x_i−1; then C(·,·).

Same solve path

  1. Write constraints.
  2. Shift if positive.
  3. Apply stars-and-bars.
  4. Tiny analog check.

Rough skeleton

  • Constraints
  • Shift?
  • C(…)

Why faster

One shift rule covers a huge Advanced subfamily.

Refuse this

Do not use positive formula for non-negative (or reverse).

03

Gap / non-consecutive selection

Looks likeNo two adjacent; consecutive seats; gaps

Fast modelGap transform (y_i=x_i−(i−1)) → ordinary combination.

Same solve path

  1. State gap condition.
  2. Map to unrestricted choice.
  3. Compute C on reduced n.

Rough skeleton

  • Map
  • New n
  • C

Why faster

Transform is the method; arithmetic is short.

Refuse this

Do not casework seat-by-seat.

04

At least / complement

Looks likeAt least one girl; none of type A; complementary counting

Fast modelTry total − unwanted before multi-case ‘at least’ expansions.

Same solve path

  1. Define unwanted clearly.
  2. Total with same model.
  3. Subtract.
  4. Check double-subtraction.

Rough skeleton

  • Total
  • Unwanted
  • Difference

Why faster

Complement collapses many cases into one subtraction.

Refuse this

Do not open five overlapping ‘at least k’ cases first.

Hall timing

60–90s process

  1. 0–15s
    Name the object: arrange / choose / distribute / solutions of equation. Underline constraints.

    Circle words like distinct, identical, consecutive, at least, with/without replacement, order matters. If you cannot name the object, do not open nPr/nCr yet.

  2. 15–35s
    Write ONE model sentence: ‘Choose k from n with/without order, with/without replacement, constraint X’.

    This is the method. Example: ‘Positive integer solutions of x+y+z=10’ → shift yi=xi−1. Example: ‘No two consecutive’ → gap transform. Commit to one sentence.

  3. 35–75s
    Translate sentence → nPr / nCr / stars-and-bars / complement / cases. Keep exact integers.

    Split into mutually exclusive cases only when constraints force it. Prefer complement when ‘at least one’ is messy. Never change order-assumption mid-solution.

  4. 75–90s
    Overcount check: did you double-count ordered pairs? Did identical objects need division?

    Sanity: answer is a non-negative integer; compare to a smaller analogous count. For MC, reject options that ignore a stated constraint.

After the model is locked

Quick formula bank

Permutation

nPr = n!/(n−r)!

Order matters; distinct objects; no repetition unless stated

Combination

nCr = n!/(r!(n−r)!)

Order does not matter; selection only

With repetition (arrangements)

n^r

Ordered r-tuples from n types with replacement

Stars and bars (non-negative)

C(n+k−1, k−1)

x1+…+xk=n, xi≥0 integers

Stars and bars (positive)

C(n−1, k−1)

x1+…+xk=n, xi≥1 → set yi=xi−1≥0

Identical objects arrangements

n!/(n1! n2! …)

Multiset permutations

No two consecutive (classic map)

Choose k from n−k+1 after gap transform

Selecting k numbers from 1…n with gaps ≥2

Complement

Total − unwanted

‘At least one’ / ‘none of type’ is cleaner in reverse

Mental bank

  • Order matters → P; only selection → C
  • xi≥1 ⇒ yi=xi−1 then stars-and-bars
  • Identical letters → divide by repeat factorials
  • At least one → try complement first
  • nCr = nC(n−r)
  • Committee with roles → P; without roles → C
  • Gap transform before ‘no two consecutive’ combinations
  • Complement: total − forbidden often beats nested ‘at least’ cases
  • Circular arrangements: (n−1)! for distinct seats up to rotation

Make it automatic

Practice · checklist

Daily

  • 3 timed counts (≤90s each): write model sentence by 20s on every stem.
  • 1 stars-and-bars (rotate ≥0 vs ≥1).
  • Log any stem where formula was chosen before the sentence.

Weekly

  • 1 mixed set: arrangements + selections + distributions (8 questions).
  • 1 probability join where the hard part is the counting model.
  • Error log: P↔C swaps and missed shifts — one anti-trap line each.

100% checklist

  • Every stem has a written model sentence before nPr/nCr.
  • Positivity/non-negativity stated before stars-and-bars.
  • Identical objects always trigger multiset division check.
  • Complement considered for ‘at least’ wording.
  • Final overcount check in the last 15s.
  • Accuracy ≥90% on last timed P&C set.
  • Model sentence includes identical vs distinct and order yes/no.
  • Complement considered whenever ‘at least’ appears.
  • Case splits checked for mutual exclusion before adding.
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