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Permutations & Combinations revision notes

Every idea in the Permutations & Combinations chapter for JEE Main, with the usual mistakes and the fast methods. 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 concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Counting principles and factorials

Multiplication and addition rules
5× asked, last 2025
AND (successive steps)⇒m×n,OR (separate cases)⇒m+n\text{AND (successive steps)} \Rightarrow m\times n,\qquad \text{OR (separate cases)} \Rightarrow m+n
Multiply for independent successive steps; 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}^{n} k\cdot k! = (n+1)!-1
n!n! is defined for whole numbers n≥0n\ge0. The sum telescopes because k⋅k!=(k+1)!−k!k\cdot k!=(k+1)!-k!.
Counting functions
4× asked, last 2025
nm (all),nPm (one-one),n! (bijections, m=n)n^{m}\ \text{(all)},\qquad {}^{n}P_{m}\ \text{(one-one)},\qquad n!\ \text{(bijections, } m=n)
f:A→Bf:A\to B with ∣A∣=m|A|=m, ∣B∣=n|B|=n: each input picks an output. One-one needs m≤nm\le n.
  • Not counting equal values for a one-one ff. A condition like f(1)+f(3)=14f(1)+f(3)=14 cannot use 7+77+7. List pairs of distinct values (each in both orders), then fill the other m−2m-2 inputs from the n−2n-2 unused values: n−2Pm−2{}^{n-2}P_{m-2}.
Complement and inclusion–exclusion
9× asked, last 2025
n(A∪B)=n(A)+n(B)−n(A∩B),n(neither)=N−n(A∪B)n(A\cup B)=n(A)+n(B)-n(A\cap B),\qquad n(\text{neither})=N-n(A\cup B)
For "at least one", "neither … nor", "not all". Three sets: subtract pairs, add the triple back.
  • Multiples of kk in [a,b][a,b]: ⌊b/k⌋−⌊(a−1)/k⌋\lfloor b/k\rfloor-\lfloor (a-1)/k\rfloor. Divisible by 3 or 11: add both counts, subtract the multiples of 33.
  • Derangements (none in its own place): Dn=n!∑k=0n(−1)k/k!D_n=n!\sum_{k=0}^{n}(-1)^k/k!; D3=2D_3=2, D4=9D_4=9, D5=44D_5=44. Exactly kk of nn in place: (nk)Dn−k\binom nk D_{n-k}.
  • Forbidden strings of nn distinct letters: n!−N(A)−N(B)+N(A∩B)n!-N(A)-N(B)+N(A\cap B), each string tied into one unit. Strings sharing a letter: merge if one ends where the other begins (AB, BC: ABC), else N(A∩B)=0N(A\cap B)=0.
Remainder and gcd conditions
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
Pull dd out, then count coprime pairs in the new range. Gcd with a fixed number: decide prime by prime.
  • gcd⁡(n,54)=2\gcd(n,54)=2 means nn even but not a multiple of 3: even 4-digit numbers minus multiples of 6, 4500−1500=30004500-1500=3000. For 36 also exclude multiples of 4.
  • Quick method. Sum conditions mod kk: sort numbers by remainder. x≠yx\ne y from 1–25 with 5∣x+y5\mid x+y: classes (0,0),(1,4),(2,3)(0,0),(1,4),(2,3) give 5⋅4+2⋅25+2⋅25=1205\cdot4+2\cdot25+2\cdot25=120 ordered pairs.

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Practise Permutations & Combinations

27 more concepts in 6 topics

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

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Permutations

  • Permutations of distinct objects4×
  • Repetition allowed6×
  • Identical objects in a row7×
  • r-letter words from a word with repeats6×
  • Digit problems and divisibility19×

Restricted and circular arrangements

  • String method (kept together)4×
  • Not all together (complement)5×
  • Gap method (no two together)2×
  • Circular arrangements1×

Rank of a word

  • Rank with distinct letters7×
  • Rank with repeated letters6×
  • Word at a given position6×

Combinations and selections

  • Combinations4×
  • Properties of nCr6×
  • At least / at most selections13×
  • Selecting from identical objects1×
  • Subsets and forced order8×
  • Points, lines and triangles6×
  • Regular polygon counts4×
  • Grid paths and first-to-n series1×

Distribution and integer solutions

  • Identical objects, distinct boxes (beggar's)10×
  • Upper bounds on each share6×
  • Fixed product: share the exponents5×
  • Distinct objects, distinct boxes4×
  • Division into groups4×
  • Identical boxes2×

Prime powers in n!

  • Prime power in n! (Legendre)2×

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