Permutations and Combinations Class 11 Notes Maths Chapter 6

 Page 1


 
KEY POINTS
? Multiplication Principle (Fundamental Principle of Counting: 
If an event can occur in m different ways, following which 
another event can occur in n different ways, then the total no. of 
different ways of occurrence of the two events in order is m × n. 
? Fundamental Principle of Addition:If there are two events 
such that they can occur independently in m and n different 
ways respectively, then either of the two events can occur in (m 
+ n) ways. 
? Factorial:Factorial of a natural number n, denoted by n! or n is 
the continued product of first n natural numbers. 
n! = n × (n – 1) × (n – 2) × ... × 3 × 2 × 1 
 = n × ((n – 1)!) 
 = n × (n – 1) × ((n – 2)!) 
? Permutation:A permutation is an arrangement of a number of 
objects in a definite order taken some or all at a time. 
? The number of permutation of n different objects taken r at a 
time where0 ? r ? n and the objects do not repeat is denoted by 
n
Pror P(n, r) where, 
Page 2


 
KEY POINTS
? Multiplication Principle (Fundamental Principle of Counting: 
If an event can occur in m different ways, following which 
another event can occur in n different ways, then the total no. of 
different ways of occurrence of the two events in order is m × n. 
? Fundamental Principle of Addition:If there are two events 
such that they can occur independently in m and n different 
ways respectively, then either of the two events can occur in (m 
+ n) ways. 
? Factorial:Factorial of a natural number n, denoted by n! or n is 
the continued product of first n natural numbers. 
n! = n × (n – 1) × (n – 2) × ... × 3 × 2 × 1 
 = n × ((n – 1)!) 
 = n × (n – 1) × ((n – 2)!) 
? Permutation:A permutation is an arrangement of a number of 
objects in a definite order taken some or all at a time. 
? The number of permutation of n different objects taken r at a 
time where0 ? r ? n and the objects do not repeat is denoted by 
n
Pror P(n, r) where, 
 
!
( )!
n
r
n
P
n r
?
?
? The number of permutations of n objects, taken r at a time, when 
repetition of objects is allowed is n
r
. 
? The number of permutations of n objects of which p1 are of one 
kind, p2 are of second kind, …….. pk are of k
th
 kind and the rest 
if any, are of different kinds, is
1 2
!
! !....... !
k
n
p p p
? Combination:Each of the different selections made by choosing 
some or all of a number of objects, without considering their 
order is called a combination. The number of combination of n 
objects taken r at a time where 
0 ? r ? n, is denoted by 
n
Cr or C(n, r) or
n
r
? ?
? ?
? ?
 where 
n
Cr
!
!( )!
n
r n r
?
?
Some important result : 
0! = 1 
n
C0 = 
n
Cn =  1 
n
Cr = 
n
Cn–r where 0 ? r ? n, and r are positive integers 
n
Pr = n
n
Cr where 0 ? r ? n, r and n are positive integers. 
n
Cr + 
n
Cr+1 = 
n+1
Cr+1 where 0 ? r ? n and r and N are positive 
integers. 
If
n
Ca =
 n
Cb if either a = b or a + b = n