Fundamental Principle of Counting
Multiplication Principle
If one job → m ways
Second job (after first) → n ways
⇒ Both jobs together → m × n ways
Addition Principle
Job A → m ways
Job B → n ways
⇒ Either job → m + n ways
Permutations (Order Matters)
^nP_r = n! / (n-r)! = n(n-1)(n-2)...(n-r+1)
^nP_n = n! ^nP_0 = 1 ^nP_1 = n
^nP_n = n! ^nP_0 = 1 ^nP_1 = n
- ^nP_r = n × (^(n-1)P_(r-1))
- Repetition allowed → n^r
- Not all distinct (p alike of 1 kind, q of 2nd, ...) → n! / (p! q! ...)
Special Cases of Permutations
Repetition Allowed
n distinct objects, taken r at a time
→ n^r
Not All Distinct
p alike, q alike, ... rest distinct
→ n! / (p! q! ...)
Circular Permutations
n distinct objects in circle → (n–1)!
Taken r at a time:
Clockwise ≠ Anticlockwise → ^nP_r
Clockwise = Anticlockwise → ^nP_r / 2
Taken r at a time:
Clockwise ≠ Anticlockwise → ^nP_r
Clockwise = Anticlockwise → ^nP_r / 2
Combinations (Order Doesn't Matter)
^nC_r = n! / (r!(n-r)!) = ^nP_r / r!
^nC_0 = ^nC_n = 1 ^nC_1 = n ^nC_r = ^nC_(n-r)
^nC_0 = ^nC_n = 1 ^nC_1 = n ^nC_r = ^nC_(n-r)
Maximum value of ^nC_r:
- n even → ^nC_(n/2)
- n odd → ^nC_((n+1)/2) or ^nC_((n-1)/2)
Selection Cases
Distinct Objects (At least one)
^nC_1 + ^nC_2 + ... + ^nC_n = 2^n – 1
Identical Objects (At least one)
a1 alike, a2 alike, ..., k distinct
→ (a1+1)(a2+1)...(an+1) × 2^k – 1
Division & Grouping
Distinct objects into distinct groups (m+n+p things)
^(m+n+p)C_m × ^(n+p)C_n × ^pC_p = (m+n+p)! / (m! n! p!)
mn distinct things → m equal groups (order matters)
(mn)! / (n!)^m
n identical things → r groups
Empty allowed → ^(n+r-1)C_(r-1)
Empty not allowed → ^(n-1)C_(r-1)
n distinct things → r groups (arrangement)
Empty allowed → n! × ^(n+r-1)C_(r-1)
Empty not allowed → n! × ^(n-1)C_(r-1)
JEE Main Tips & Weightage
One of the highest scoring & most frequent chapters – 4–6 questions almost every year!
Must Remember for JEE
- (n–1)! for circular
- n^r when repetition allowed
- Divide by factorial for identical objects
- 2^n – 1 for “at least one” from distinct
- Division of identical → stars & bars ^(n+r-1)C_(r-1)
- Circular with same clockwise/anticlockwise → divide by 2
- Maximum binomial coefficient