JEE JEE Main and JEE Advanced · Mathematics

Permutations and Combinations

Model a counting problem correctly before choosing a formula by deciding whether order matters, whether repetition is allowed, whether objects are distinguishable, and what restrictions apply.

Subject
Mathematics
Syllabus unit
Permutations and Combinations
  • Build the counting model before picking a formula
  • Order, repetition and distinguishability decide the formula
  • No invented weightage, question counts or trend percentages

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In short

Permutation and combination formulas are outputs of a counting model, not the starting point. First decide what is being counted.

If order matters, use an arrangement model. If order does not matter, use a selection model. Then decide whether repetition is allowed, whether any objects are indistinguishable, and whether restrictions split the problem into cases.

Syllabus mapping

  • Unit
    Permutations and Combinations
    Topics
    Fundamental principle of counting, Permutations, Combinations, Meaning of P(n,r) and C(n,r), Simple applications

What this chapter contains and why it matters

  • Question
    What is the chapter about?
    Direct answer
    Building a counting model, then choosing between permutation, combination or a supporting counting formula.
  • Question
    What is the central method choice?
    Direct answer
    Define the outcome, decide order, repetition and distinguishability, apply restrictions, then select the formula.
  • Question
    Where do most mistakes begin?
    Direct answer
    Choosing a formula before defining the counted outcome, and treating identical objects as distinguishable.
  • Question
    What should come before this chapter?
    Direct answer
    Sets and Relations, for set-based case distinction.
  • Question
    What comes after it?
    Direct answer
    Binomial Theorem uses combinations notation; Probability builds on counting models.

The official JEE documents define content scope. They do not publish chapter weightage, so none is asserted here.

Official JEE syllabus mapping for Permutations and Combinations

Verified against the current JEE Main 2026 syllabus and JEE Advanced 2026 syllabus.

  • Concept group
    Counting principle and formulas
    JEE Main 2026
    Unit 4 explicitly includes the fundamental principle of counting, permutations and combinations, the meaning of P(n,r) and C(n,r), and simple applications.
    JEE Advanced 2026
    Algebra explicitly includes permutations and combinations.

Sources: JEE Main 2026 syllabus and JEE Advanced 2026 syllabus, both linked in the sources section below.

Before this chapter

Concepts in this chapter

1. Define the counted outcome in words

State in words what one counted outcome is before choosing any formula.

State in words what one counted outcome is. This prevents jumping to a formula before the model is clear.

2. Decide whether order matters

Order determines whether the model is a permutation or a combination.

Decide whether changing order creates a different outcome. If it does, use a permutation-based model; if not, use a combination-based model.

3. Decide whether repetition is allowed

Repetition changes the formula from factorial-based to power-based counting.

Decide whether an option may be reused. Do not assume repetition is allowed simply because a formula would be convenient.

4. Identify identical objects

Interchanging identical objects does not create a new outcome.

Identify identical objects before counting arrangements. Treating identical objects as distinguishable overcounts the result.

5. Apply restrictions by cases, complement or position fixing

Restrictions can split a problem into disjoint cases or be handled through complements.

Apply restrictions by cases, complement or position fixing, then only select nPr, nCr, factorial division or a product of choices once the model is settled.

Method selector

  • Question signal
    Order matters, no repetition
    First model
    Permutation.
  • Question signal
    Order does not matter, no repetition
    First model
    Combination.
  • Question signal
    Independent labelled positions, repetition allowed
    First model
    Multiplication principle, often n^r.
  • Question signal
    Some objects identical
    First model
    Start from labelled arrangements, then divide by internal identical permutations when valid.
  • Question signal
    "At least one" restriction
    First model
    Complement may be simpler.
  • Question signal
    Mutually exclusive structural cases
    First model
    Count each case then add.
  • Question signal
    Object fixed together/apart
    First model
    Decide whether block or complement reasoning is valid before calculating.

Formula sheet

  • Total outcomes equal the product of the choice counts at each stage.

    Count of outcomes for a staged process with successive independent choices.

    m_i
    number of allowed choices at stage i

    Use when — Sequential choices.

    Common trap — Multiplying mutually exclusive alternatives.

  • Add the counts of cases that do not overlap.

    Add counts of disjoint cases.

    Use when — Either/or cases.

    Common trap — Double-counting overlap.

  • n factorial equals n times n minus one and so on down to one; zero factorial equals one.

    Number of ways to arrange n distinct objects.

    n
    nonnegative integer

    Use when — Arrange n distinct objects.

    Common trap — Undefined negative factorial.

  • n P r equals n factorial divided by n minus r factorial.

    Number of ordered arrangements of r objects chosen from n distinct objects, no repetition.

    n
    distinct available objects
    r
    integer, 0≤r≤n

    Use when — Arrange r from n.

    Common trap — Using when order does not matter.

  • n C r equals n factorial divided by r factorial times n minus r factorial.

    Number of unordered selections of r objects from n distinct objects, no repetition.

    n
    distinct objects
    r
    integer, 0≤r≤n

    Use when — Choose r from n.

    Common trap — Counting order repeatedly.

  • n P r equals n C r times r factorial.

    Relation between ordered and unordered counts.

    Use when — Convert selection then arrangement.

    Common trap — Forgetting r!.

  • n to the power r.

    Count of ordered sequences of r labelled positions, each with n available choices, repetition allowed.

    n
    choices per position
    r
    number of positions

    Use when — Codes or sequences under stated rules.

    Common trap — Using when restrictions change by position.

  • n factorial divided by the product of the factorials of each identical group's size.

    Distinct arrangements of n objects with repeated identical groups.

    m_i
    size of identical group i, summing to n

    Use when — Repeated identical objects.

    Common trap — Dividing for objects that are actually distinguishable.

Worked examples

How many distinct arrangements can be made from the letters of LEVEL?

Answer: 30

Five letters total. L appears twice, E appears twice, V appears once. Order matters, but the L copies and E copies are indistinguishable.

If all five positions contained labelled objects there would be 5! arrangements. Each visible arrangement is counted 2! times for the L copies and 2! times for the E copies.

5!/(2!2!) = 120/4 = 30.

Validity check: the result is smaller than 5!=120, as it must be because identical-letter swaps do not create new outcomes.

Common mistakes and what they actually indicate

  • Choosing a formula before defining the counted outcome.

    Decision / selection error

    Why it happens

    Without a defined outcome, it is unclear whether order, repetition or distinguishability apply.

    How it is corrected

    State in words what one counted outcome is before selecting a formula.

  • Using permutation when order does not matter.

    Decision / selection error

    Why it happens

    This overcounts outcomes that are actually the same selection.

    How it is corrected

    Confirm whether order changes the outcome before choosing nPr or nCr.

  • Using combination when order does matter.

    Decision / selection error

    Why it happens

    This undercounts outcomes that should be treated as distinct.

    How it is corrected

    Confirm whether order changes the outcome before choosing nPr or nCr.

  • Assuming repetition is allowed because the formula is convenient.

    Needs review

    Why it happens

    Repetition must be justified by the problem statement, not assumed.

    How it is corrected

    Check the problem wording explicitly for repetition allowance.

  • Treating identical objects as distinguishable.

    Execution error

    Why it happens

    This overcounts arrangements by the number of internal permutations of the identical group.

    How it is corrected

    Identify identical objects and divide by their internal factorial.

  • Adding overlapping cases.

    Execution error

    Why it happens

    The addition principle requires disjoint cases.

    How it is corrected

    Check for intersections between cases before adding their counts.

  • Multiplying mutually exclusive alternatives.

    Decision / selection error

    Why it happens

    Multiplication applies to sequential independent choices, not either/or alternatives.

    How it is corrected

    Use the addition principle for either/or alternatives instead.

  • Double-counting when restrictions create overlapping cases.

    Needs review

    Why it happens

    Restriction cases that overlap will be counted more than once if added directly.

    How it is corrected

    Verify restriction cases are disjoint, or use complement reasoning instead.

Sources and provenance

Official syllabus scope verified against NTA JEE Main 2026 and JEE Advanced 2026 documents. No weightage, frequency or PYQ-count claim is made.

Contributor requirements for this page

  • Reviewer specialisation: combinatorics and probability foundations.
  • Minimum: postgraduate Mathematics, Applied Mathematics or equivalent.
  • Review scope: model conditions for every count, overlap logic, repetition assumptions, indistinguishable-object formula and worked example.