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Permutations & Combinations formula sheet

Every formula in the Permutations & Combinations chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 32 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that formula, and "last" is the latest year one appeared. ★ marks formulas asked 8 or more times.

Counting principles and factorials

Multiplication and addition
5× asked, last 2025
AND (successive steps)⇒m×nOR (separate cases)⇒m+n\begin{aligned}&\text{AND (successive steps)}\Rightarrow m\times n\\&\text{OR (separate cases)}\Rightarrow m+n\end{aligned}
Add only for cases that cannot happen together.
Factorial
1× asked, last 2021
n!=n (n−1)!,0!=1,∑k=1nk⋅k!=(n+1)!−1n!=n\,(n-1)!,\qquad 0!=1,\qquad \sum_{k=1}^nk\cdot k!=(n+1)!-1
Telescopes: k⋅k!=(k+1)!−k!k\cdot k!=(k+1)!-k!.
Counting functions
4× asked, last 2025
nm (all),nPm (one-one),n! (bijections)n^m\ (\text{all}),\qquad {}^nP_m\ (\text{one-one}),\qquad n!\ (\text{bijections})
∣A∣=m|A|=m, ∣B∣=n|B|=n. One-one: no equal values; fill the remaining inputs with n−kPm−k{}^{n-k}P_{m-k}.
Inclusion–exclusion and derangements
9× asked, last 2025
n(A∪B)=n(A)+n(B)−n(A∩B)Dn=n!∑k=0n(−1)kk!:D3=2, D4=9, D5=44\begin{aligned}&n(A\cup B)=n(A)+n(B)-n(A\cap B)\\&D_n=n!\sum_{k=0}^n\frac{(-1)^k}{k!}:\quad D_3=2,\ D_4=9,\ D_5=44\end{aligned}
Multiples of kk in [a,b][a,b]: ⌊b/k⌋−⌊(a−1)/k⌋\lfloor b/k\rfloor-\lfloor(a-1)/k\rfloor. Exactly kk in place: nCk Dn−k{}^nC_k\,D_{n-k}.
gcd and remainders
6× asked, last 2026
gcd⁡(m,n)=d  ⟺  m=da, n=db, gcd⁡(a,b)=1\gcd(m,n)=d\iff m=da,\ n=db,\ \gcd(a,b)=1
Gcd with a fixed number: prime by prime. Sum conditions mod kk: sort numbers by remainder class.

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Practise Permutations & Combinations

27 more formulas in 5 topics

Permutations, Restricted and circular arrangements, Rank of a word, Combinations and selections and Distribution and integer solutions. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Permutations

  • Distinct objects4×
  • Repetition allowed6×
  • Identical objects7×
  • r-letter words with repeats6×
  • Digits and divisibility19×

Restricted and circular arrangements

  • Kept together (string)4×
  • Not all together5×
  • No two together (gaps)2×
  • Circular arrangements1×

Rank of a word

  • Rank, distinct letters7×
  • Rank, repeated letters6×
  • Word at a position6×

Combinations and selections

  • Combinations4×
  • Properties of nCr6×
  • At least / at most13×
  • Selecting from identical objects1×
  • Subsets and forced order8×
  • Points, lines, triangles6×
  • Regular polygon4×
  • Grid paths and rectangles1×

Distribution and integer solutions

  • Identical into distinct (beggar)10×
  • Upper bounds6×
  • Fixed product5×
  • Distinct into distinct4×
  • Division into groups4×
  • Identical boxes2×
  • Legendre (prime in n!)2×

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Practise Permutations & Combinations

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