Permutations and Combinations
Permutations And Combinations PDF Notes, Important Questions and Synopsis
SYNOPSIS
 Fundamental Principle of Counting
 Permutation is the number of ways to arrange things.
Eg: Password is 2045
(order matters)  It is denoted by P(n, r) and given by
P(n, r) =, where 0 ≤ r ≤ n
n → number of things to choose from
r → number of things we choose
! → factorial
 Combination is the number of ways to choose things.Eg: A cake contains chocolates, biscuits, oranges and cookies.
(Order does not matter)  It is denoted by C(n, r) and given by
C(n, r) =, where 0 ≤ r ≤
n → number of things to choose from
r → number of things we choose
! → factorial
 Permutation is the number of ways to arrange things.
 Permutation
If n is the number of distinct things and r things are chosen at a time.
i. Permutation of objects when all are not distinct:
Permutation = , → Number of things among ‘n’ which are alike of rth type.
ii. Circular permutation
1. When clockwise and anticlockwise arrangements are different:
Number of permutations: (n  1)!
2. When clockwise and anticlockwise arrangements are the same:
Number of permutations:
iii. Permutation under restrictions
Selecting and arranging r distinct objects from n
 When ‘k’ particular things are always to be included.
Number of permutations:  When a particular thing is always to be included (k = 1).
Number of permutations:  When ‘k’ particular things are never included.
Number of permutations:  When a particular thing is never included.
Number of permutations:  When ‘l’ particular things always come together.
Number of permutations: (n  l + 1)! ⨯ l!  When ‘l’ particular things never come together.
Number of permutations: n!  (n  l + 1)! ⨯ l!
i. Permutations with repetition Number of permutations: ^{n}P_{r}= n^{r} (Repetition, order matters)
Out of ← Taking 2 at a time 
ii. Permutations without repetition
(No repetition, order matters) Out of ← Taking 2 at a time 
3.Combination


ii Combinations under restriction


i. Combination with repetition Formula: (Repetition, order does not matter) Out of _{ }← Taking 2 at a time 
ii. Combination without repetition Formula: (No repetition, order does not matter) Out of ←Taking 2 at a time 
Download complete content for FREE
JEE Main  Maths
Asked by shivanshij5  07 Feb, 2024, 11:16: AM
ANSWERED BY EXPERT
JEE Main  Maths
Asked by pantsanjana10  28 Jan, 2024, 09:10: AM
ANSWERED BY EXPERT
JEE Main  Maths
Asked by rekha.rmd7  12 Dec, 2023, 12:04: AM
ANSWERED BY EXPERT
JEE Main  Maths
Asked by harshpunia109  26 Jun, 2023, 07:21: PM
ANSWERED BY EXPERT
JEE Main  Maths
Asked by harshpunia109  25 Jun, 2023, 06:36: PM
ANSWERED BY EXPERT
JEE Main  Maths
Asked by abhishekssstays  17 Dec, 2022, 05:00: PM
ANSWERED BY EXPERT
JEE Main  Maths
Asked by amadhusudhan09  28 Oct, 2021, 07:20: PM
ANSWERED BY EXPERT
Related Chapters
 Sets, Relations and Functions
 Complex Numbers and Quadratic Equations
 Matrices and Determinants
 Mathematical Induction
 Binomial Theorem and its Simple Applications
 Sequences and Series
 Limit, Continuity and Differentiability
 Integral Calculus
 Differential Equations
 Coordinate Geometry
 Three Dimensional Geometry
 Vector Algebra
 Statistics and Probability
 Trigonometry
 Mathematical Reasoning