Math · Algebra

Permutations and Combinations formulas for JEE

Every Permutations and Combinations formula you need for JEE, grouped by concept.

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21 formulas2 concepts
01Fundamental Principle of Counting02Permutations and Combinations
01

Fundamental Principle of Counting

5 formulas

Factorial Notation

n!=1×2×3×⋯×(n−1)×nn! = 1 \times 2 \times 3 \times \dots \times (n-1) \times n

Product of the first n natural numbers.

applies whenn≥1,n∈Nn \ge 1, n \in \mathbb{N}
factorialnotation

Recursive Property of Factorial

n!=n(n−1)!n! = n(n-1)!

Relationship between factorial of n and factorial of (n-1).

applies whenn≥1n \ge 1
factorialrecursive

Zero Factorial

0!=10! = 1

By definition, the number of ways to arrange zero objects.

factorialzero_factorial

Legendre's Formula

Ep(n!)=∑k=1∞⌊npk⌋E_p(n!) = \sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloor

Finds the highest power (exponent) of a prime number p that divides n!.

applies whenp is a prime number.
factorialprime_factorizationlegendrejee-advanced

Fundamental Principle of Counting

m×n×p…m \times n \times p \dots

Total number of ways sequential independent events can occur.

applies whenEvents are independent and occur in a definite order.
multiplication_principlecounting
02

Permutations and Combinations

16 formulas

Distribution of Identical Objects (Non-Negative)

n+r−1Cr−1^{n+r-1}C_{r-1}

Number of non-negative integral solutions to x_1 + x_2 + ... + x_r = n. (Beggars Method / Stars and Bars).

applies whenxi≥0x_i \ge 0
combinationsdistributionstars_and_barsjee-advanced

Distribution of Identical Objects (Positive)

n−1Cr−1^{n-1}C_{r-1}

Number of strictly positive integral solutions to x_1 + x_2 + ... + x_r = n.

applies whenxi≥1,n≥rx_i \ge 1, n \ge r
combinationsdistributionstars_and_barsjee-advanced

Circular Permutations

(n−1)!(n-1)!

Number of ways to arrange n distinct objects in a circle.

applies whenClockwise and counter-clockwise arrangements are distinct.
permutationscircularjee-advanced

Circular Permutations (Symmetric)

(n−1)!2\frac{(n-1)!}{2}

Number of ways to arrange n distinct objects in a circle when clockwise and counter-clockwise are indistinguishable.

applies whenE.g., beads on a necklace, flowers in a garland.
permutationscircularsymmetricjee-advanced

Derangements

Dn=n!∑k=0n(−1)kk!D_n = n! \sum_{k=0}^{n} \frac{(-1)^k}{k!}

Number of permutations where no element appears in its original position.

permutationsderangementsjee-advanced

Distribution into Identical Groups

n!(p!)kk!\frac{n!}{(p!)^k k!}

Number of ways to distribute n objects into k identical groups of size p (where n = pk).

applies whenGroups are unmarked/indistinguishable.
permutationsdistributiongroupingjee-advanced

Permutations with Identical Objects

n!p1!p2!…pk!\frac{n!}{p_1! p_2! \dots p_k!}

Permutations of n objects where p1, p2, ... pk objects are of the same kind respectively.

applies whenp1+p2+⋯+pk≤np_1 + p_2 + \dots + p_k \le n
permutationsidentical_objects

Combination Formula

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

Number of combinations (selections) of n distinct objects taken r at a time.

applies when0≤r≤n0 \le r \le n
combinationsselection

Combinations Equality Rule

nCa=nCb  ⟹  a=b or a+b=n^nC_a = ^nC_b \implies a = b \text{ or } a + b = n

Equality condition for two combinations from the same set of n items.

applies whena,b≥0a, b \ge 0
combinationsequalityidentities

Combination Symmetry

nCr=nCn−r^nC_r = ^nC_{n-r}

Selecting r objects is mathematically equivalent to leaving behind (n-r) objects.

applies when0≤r≤n0 \le r \le n
combinationssymmetry

Permutations of All Objects

nPn=n!^nP_n = n!

Arranging n distinct objects taken all at a time.

permutationsarrangementall_objects

Permutation Formula

nPr=n!(n−r)!^nP_r = \frac{n!}{(n-r)!}

Number of permutations of n distinct objects taken r at a time without repetition.

applies when0≤r≤n0 \le r \le n
permutationsarrangementdistinct

Pascal's Rule

nCr+nCr−1=n+1Cr^nC_r + ^nC_{r-1} = ^{n+1}C_r

Sum of two adjacent combinations in Pascal's triangle.

applies when1≤r≤n1 \le r \le n
combinationspascalidentities

Relation between nPr and nCr

nPr=nCr×r!^nP_r = ^nC_r \times r!

Relates arrangements and selections: select r objects, then arrange them.

applies when0≤r≤n0 \le r \le n
permutationscombinationsrelationship

Permutations with Repetition

nrn^r

Number of permutations of n distinct objects taken r at a time with repetition allowed.

permutationsrepetition

Total Combinations (At Least One)

2n−12^n - 1

Total number of ways to select at least one object from n distinct objects.

combinationssubsetsjee-advanced
Other chapters
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