Mathematics · JEE

Permutations and Combinations Formula Sheet for JEE

45+ JEE formulas in this unit

Quick answer

The Permutations and Combinations JEE formula sheet lists 45+ important formulas for JEE Main and Advanced, including essential identities from The fundamental principle of counting, Permutations and combinations, Meaning of P(n,r) and C(n,r), Simple applications. Revise essential formulas first, then practise MCQs on Goodmarks.

Download-free JEE mathematics formula revision for Permutations and Combinations. This unit-wise formula list covers 45+ exam-relevant results across The fundamental principle of counting, Permutations and combinations, Meaning of P(n,r) and C(n,r), Simple applications, organised by subtopic for quick last-minute revision.

JEE Formula Sheet

45 formulas across 4 subtopics — organised for JEE Main & Advanced revision

Practise MCQs for this unit

Essential: 18Important: 22Supplementary: 5

The fundamental principle of counting

Addition principle of countingEssential

\text{If one task can be done in }m\text{ ways and another disjoint task in }n\text{ ways, then one of them can be done in }m+n\text{ ways.}

Variables: $m,n$ are the numbers of ways of mutually exclusive choices.
Conditions: Choices counted must be mutually exclusive and collectively represent the required selection.
Where used in JEE: Counting cases split into non-overlapping categories; arrangement/selection by cases.

Multiplication principle of countingEssential

\text{If a process has successive independent stages with }n_1,n_2,\dots,n_k\text{ choices respectively, then total ways }=n_1n_2\cdots n_k.

Variables: $n_1,n_2,\dots,n_k$ are numbers of choices at successive stages.
Conditions: Each final outcome is formed by choosing one option from each stage.
Where used in JEE: Core counting in passwords, digit arrangements, distributions, constrained permutations.

Permutations and combinations

Factorial definitionEssential

n!=1\cdot 2\cdot 3\cdots n

for

n\ge 1

, and

0!=1

Variables: $n$ is a non-negative integer.
Conditions: Defined for non-negative integers.
Where used in JEE: Permutations, combinations, multinomial counts, circular arrangements.

Factorial recurrenceImportant

(n+1)!=(n+1)n!

Variables: $n$ is a non-negative integer.
Conditions: Valid for all non-negative integers $n$ .
Where used in JEE: Simplifying permutation/combination expressions and recurrences.

Symmetry of combinationsEssential

\,{}^nC_r={}^nC_{n-r}

Variables: $n$ is total number of objects, $r$ is number selected.
Conditions: $0\le r\le n$ .
Where used in JEE: Simplification, finding greatest binomial coefficient, complementary counting.

Boundary values of combinationsImportant

\,{}^nC_0={}^nC_n=1,\quad {}^nC_1={}^nC_{n-1}=n

Variables: $n$ is a non-negative integer.
Conditions: Valid for $n\ge 1$ where needed.
Where used in JEE: Edge cases in identities and expansions.

Pascal identityEssential

\,{}^nC_r={} ^{n-1}C_r+{}^{n-1}C_{r-1}

Variables: $n,r$ are integers.
Conditions: $1\le r\le n-1$ .
Where used in JEE: Recursive evaluation, binomial coefficient proofs, counting by inclusion of a fixed element.

Ratio of consecutive combinationsImportant

\dfrac{{}^nC_{r+1}}{{}^nC_r}=\dfrac{n-r}{r+1}

Variables: $n$ is fixed, $r$ varies.
Conditions: $0\le r\le n-1$ .
Where used in JEE: Finding greatest term/coefficient, monotonicity of binomial coefficients.

Useful combination recurrence 1Important

\,{}^nC_r=\dfrac{n}{r}\,{}^{n-1}C_{r-1}

Variables: $n,r$ are integers.
Conditions: $1\le r\le n$ .
Where used in JEE: Simplifying expressions and solving equations involving binomial coefficients.

Useful combination recurrence 2Important

\,{}^nC_r=\dfrac{n}{n-r}\,{}^{n-1}C_r

Variables: $n,r$ are integers.
Conditions: $0\le r\le n-1$ .
Where used in JEE: Simplifying expressions and comparing adjacent coefficients.

Number of permutations with repetition allowedEssential

n^r

Variables: $n$ is number of available distinct objects/symbols, $r$ is number of positions filled.
Conditions: Each of the $r$ positions can be filled independently by any of the $n$ objects; repetition allowed.
Where used in JEE: Code formation, digit strings, selections with replacement where order matters.

Permutations of a multisetEssential

\frac{n!}{p_1!p_2!\cdots p_k!}

Variables: Among $n$ total objects, identical objects occur in groups of sizes $p_1,p_2,\dots,p_k$ with $p_1+\cdots+p_k=n$ .
Conditions: Objects within each group are identical.
Where used in JEE: Arrangements of words with repeated letters, repeated symbols in linear order.

Number of combinations with repetition allowedImportant

{}^{n+r-1}C_r=\frac{(n+r-1)!}{r!(n-1)!}

Variables: $n$ is number of distinct types, $r$ is number of selections.
Conditions: $n\ge 1, r\ge 0$ ; repetition allowed; order not important.
Where used in JEE: Selection of items where repeats are allowed, stars and bars basic form.

Total number of subsets of a set with $n$ elementsImportant

2^n

Variables: $n$ is the number of distinct elements.
Conditions: Includes empty set and full set.
Where used in JEE: Subset counting, case-based counting using combinations.

Sum of all combinationsEssential

\displaystyle \sum_{r=0}^{n} {}^nC_r=2^n

Variables: $n$ is a non-negative integer.
Conditions: Valid for all non-negative integers $n$ .
Where used in JEE: Subset counting, simplifying combinatorial sums.

Alternating sum of combinationsImportant

\displaystyle \sum_{r=0}^{n} (-1)^r {}^nC_r=0

Variables: $n$ is a positive integer.
Conditions: For $n\ge 1$ .
Where used in JEE: Binomial sums and alternating counting arguments.

Sum of combinations with weights $r$Important

\displaystyle \sum_{r=0}^{n} r\,{}^nC_r=n2^{n-1}

Variables: $n$ is a non-negative integer.
Conditions: Valid for all non-negative integers $n$ .
Where used in JEE: Expected-value style counting, combinatorial sum simplification.

Sum of combinations with weights $r(r-1)$Supplementary

\displaystyle \sum_{r=0}^{n} r(r-1)\,{}^nC_r=n(n-1)2^{n-2}

Variables: $n$ is a non-negative integer.
Conditions: Valid for $n\ge 2$ .
Where used in JEE: Higher weighted binomial sums.

Greatest binomial coefficientImportant

\text{Greatest value among }{}^nC_0,{}^nC_1,\dots,{}^nC_n\text{ occurs at }r=\lfloor n/2\rfloor\text{ and also at }r=\lceil n/2\rceil\text{ if }n\text{ is odd.}

Variables: $n$ is a non-negative integer.
Conditions: For even $n$ , unique greatest coefficient is ${}^nC_{n/2}$ ; for odd $n$ , two equal greatest coefficients are ${}^nC_{(n-1)/2}$ and ${}^nC_{(n+1)/2}$ .
Where used in JEE: Maximum term/coefficient questions.

Vandermonde type identitySupplementary

\displaystyle \sum_{k=0}^{r} {}^mC_k\,{}^nC_{r-k}={} ^{m+n}C_r

Variables: $m,n,r$ are non-negative integers.
Conditions: Terms with invalid lower indices are treated as zero.
Where used in JEE: Advanced combinatorial sums and coefficient comparison.

Hockey-stick identitySupplementary

\displaystyle \sum_{k=r}^{n} {}^kC_r={} ^{n+1}C_{r+1}

Variables: $n,r$ are non-negative integers with $r\le n$ .
Conditions: Valid for integer indices in range.
Where used in JEE: Summation of diagonals in Pascal triangle.

Product identity for combinationsImportant

\,{}^nC_r\,{}^rC_k={}^nC_k\,{}^{n-k}C_{r-k}

Variables: $n,r,k$ are integers.
Conditions: $0\le k\le r\le n$ .
Where used in JEE: Two-stage selection and combinatorial simplification.

Meaning of P(n,r) and C(n,r)

Permutation of all distinct objectsEssential

^nP_n=n!

Variables: $n$ is the number of distinct objects.
Conditions: All $n$ objects are distinct and all are arranged.
Where used in JEE: Arranging all distinct objects in a line.

Permutation of $r$ objects chosen from $n$ distinct objectsEssential

{}^nP_r=\frac{n!}{(n-r)!}

Variables: $n$ is total number of distinct objects, $r$ is number arranged.
Conditions: $0\le r\le n$ , $n,r\in \mathbb{Z}_{\ge 0}$ .
Where used in JEE: Ordered selections, rank arrangements, number formation, seating.

Combination of $r$ objects chosen from $n$ distinct objectsEssential

{}^nC_r=\frac{n!}{r!(n-r)!}

Variables: $n$ is total number of distinct objects, $r$ is number selected.
Conditions: $0\le r\le n$ , $n,r\in \mathbb{Z}_{\ge 0}$ .
Where used in JEE: Unordered selections, committee formation, choosing groups.

Relation between permutation and combinationEssential

\,{}^nP_r=r!\,{}^nC_r

Variables: $n$ is total number of distinct objects, $r$ is number chosen.
Conditions: $0\le r\le n$ .
Where used in JEE: Converting ordered counts to unordered counts and vice versa.

Boundary values of permutationsImportant

\,{}^nP_0=1,\quad {}^nP_1=n

Variables: $n$ is a non-negative integer.
Conditions: Valid for non-negative integer $n$ .
Where used in JEE: Edge cases in counting and recurrence simplification.

Ratio of consecutive permutationsSupplementary

\dfrac{{}^nP_{r+1}}{{}^nP_r}=n-r

Variables: $n$ is fixed, $r$ varies.
Conditions: $0\le r\le n-1$ .
Where used in JEE: Recurrence simplification, comparison of successive permutation counts.

Useful permutation recurrence 1Important

\,{}^nP_r=n\,{}^{n-1}P_{r-1}

Variables: $n,r$ are integers.
Conditions: $1\le r\le n$ .
Where used in JEE: Recursive counting and simplification.

Useful permutation recurrence 2Important

\,{}^nP_r=(n-r+1)\,{}^nP_{r-1}

Variables: $n,r$ are integers.
Conditions: $1\le r\le n$ .
Where used in JEE: Building permutations stepwise and evaluating ratios.

Product identity involving permutationsSupplementary

\,{}^nP_r\,{}^{n-r}P_s={}^nP_{r+s}

Variables: $n,r,s$ are integers.
Conditions: $r,s\ge 0$ and $r+s\le n$ .
Where used in JEE: Sequential ordered selections without repetition.

Simple applications

Circular permutations of distinct objectsEssential

(n-1)!

Variables: $n$ is number of distinct objects.
Conditions: Objects arranged around a circle; rotations considered identical.
Where used in JEE: Round table seating, necklace-like circular seating without reflection identification.

Circular permutations when clockwise and anticlockwise arrangements are sameImportant

\dfrac{(n-1)!}{2}

Variables: $n$ is number of distinct objects.
Conditions: $n>2$ ; reflections also considered identical in addition to rotations.
Where used in JEE: Garland/necklace type arrangements where mirror images are not distinct.

Arrangements with certain objects always togetherImportant

(m!)(k!)

or more generally treat the grouped objects as one unit, so total units arranged factorially and internal arrangements multiplied separately.

Variables: $k$ objects in one group; $m$ denotes number of resulting units after grouping.
Conditions: Grouped objects are required to remain adjacent.
Where used in JEE: Seating/word arrangements with specified objects together.

Arrangements with no two specified objects togetherImportant

\text{Required count} = \text{Total arrangements} - \text{arrangements with those objects together}

Variables: Count depends on the objects and constraints of the problem.
Conditions: Use complement counting after evaluating the together-case.
Where used in JEE: Linear arrangement restrictions such as no two vowels together, men/women separation constraints.

Distribution of distinct objects into distinct boxesImportant

n^r

Variables: $n$ is number of distinct boxes, $r$ is number of distinct objects.
Conditions: Each object goes to exactly one box; boxes may be empty.
Where used in JEE: Assignment problems and mapping-style counting.

Distribution of identical objects into distinct boxes with no restrictionEssential

{}^{n+r-1}C_r={} ^{n+r-1}C_{n-1}

Variables: $r$ is number of identical objects, $n$ is number of distinct boxes.
Conditions: Boxes may be empty; each object identical.
Where used in JEE: Stars and bars, non-negative integral solutions.

Non-negative integral solutions of a linear equationEssential

x_1+x_2+\cdots+x_n=r

has

\,{}^{n+r-1}C_{r}

solutions

Variables: $x_1,x_2,\dots,x_n$ are non-negative integers.
Conditions: $r\ge 0, n\ge 1$ .
Where used in JEE: Distribution of identical objects, stars and bars, integer solution counting.

Positive integral solutions of a linear equationEssential

x_1+x_2+\cdots+x_n=r

has

\,{}^{r-1}C_{n-1}

solutions

Variables: $x_1,x_2,\dots,x_n$ are positive integers.
Conditions: $r\ge n\ge 1$ .
Where used in JEE: Distribution of identical objects with each box non-empty.

Non-negative solutions with lower boundsImportant

x_1+x_2+\cdots+x_n=r,\ x_i\ge a_i\Rightarrow

number of solutions

=\,{}^{r-\sum a_i+n-1}C_{n-1}

Variables: $a_i$ are given non-negative integers; set $y_i=x_i-a_i$ .
Conditions: $r\ge \sum a_i$ .
Where used in JEE: Integer solution problems with minimum allocation constraints.

Positive solutions with lower boundsImportant

x_1+x_2+\cdots+x_n=r,\ x_i\ge 1\Rightarrow

number of solutions

=\,{}^{r-1}C_{n-1}

Variables: $x_i$ are positive integers.
Conditions: $r\ge n$ .
Where used in JEE: At least one item per box, partitioning totals into positive parts.

Selection of at least one object from $n$ distinct objectsImportant

2^n-1

Variables: $n$ is the number of distinct objects.
Conditions: Each object is either selected or not selected.
Where used in JEE: Counting non-empty subsets, simple selection problems.

Complementary counting for selectionsEssential

\text{Required count} = \text{Total count} - \text{Unwanted count}

Variables: Counts depend on the problem context.
Conditions: Unwanted cases should be easier to count and together with wanted cases partition the sample space.
Where used in JEE: At least one, none, not together, digit/arrangement restrictions.

Arrangements of digits with first position restrictedImportant

\text{If first position has }a\text{ choices and remaining positions have }b_2,b_3,\dots,b_k\text{ choices, total}=a\,b_2b_3\cdots b_k

Variables: $a,b_2,\dots,b_k$ denote available choices at positions.
Conditions: Use position-wise counting; often first digit of a number cannot be zero.
Where used in JEE: Number formation problems from digits with/without repetition.

Choosing and arranging in two stepsEssential

\text{Number of ordered arrangements of }r\text{ chosen from }n = {}^nC_r\cdot r! = {}^nP_r

Variables: $n$ is total number of distinct objects, $r$ is number used.
Conditions: $0\le r\le n$ .
Where used in JEE: Problems naturally split into selection then arrangement.

Frequently asked questions

What are the important Permutations and Combinations formulas for JEE?

This page lists 45+ JEE-relevant Permutations and Combinations formulas organised by subtopic. Start with essential formulas, then important identities before supplementary shortcuts.

Is this Permutations and Combinations formula sheet aligned with JEE Main?

Yes. Every formula is mapped to the JEE Main Mathematics syllabus for Permutations and Combinations, covering The fundamental principle of counting, Permutations and combinations, Meaning of P(n,r) and C(n,r), Simple applications.

How should I revise the Permutations and Combinations formula sheet before JEE?

Revise essential formulas daily, important ones every 2–3 days, and supplementary results weekly. After each pass, solve 10–15 MCQs to test recall under exam conditions.

Where can I practise Permutations and Combinations MCQs after revising formulas?

Use the Online Practice or MCQs pages for the same unit on Goodmarks to convert formula recall into problem-solving speed.

Does this replace NCERT for Permutations and Combinations?

No — use this formula sheet for quick revision alongside NCERT and your coaching notes. Formulas here are a condensed reference, not a substitute for concept building.