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Permutations and Combinations Class 11 Notes Maths Chapter 6

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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 
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FAQs on Permutations and Combinations Class 11 Notes Maths Chapter 6

1. What is the difference between permutation and combination?
Ans. Permutation refers to the arrangement of objects in a particular order, while combination refers to the selection of objects without considering the order. In permutation, the order matters, whereas in combination, the order does not matter.
2. How do you calculate the number of permutations?
Ans. The number of permutations can be calculated using the formula nPr = n! / (n - r)!, where n represents the total number of objects and r represents the number of objects taken at a time. The exclamation mark denotes factorial, which means multiplying a number by all the positive integers less than it.
3. What is the formula for calculating combinations?
Ans. The formula for calculating combinations is nCr = n! / (r! * (n - r)!), where n represents the total number of objects and r represents the number of objects taken at a time. The exclamation mark denotes factorial, which means multiplying a number by all the positive integers less than it.
4. How do permutations and combinations relate to JEE?
Ans. Permutations and combinations are important topics in the JEE (Joint Entrance Examination) syllabus. Questions related to these concepts often appear in the mathematics section of the exam. Understanding permutations and combinations is essential for solving various JEE-level problems and maximizing your score in the exam.
5. Can you provide an example problem involving permutations and combinations?
Ans. Sure! Here's an example problem: How many different ways can the letters of the word "MATHS" be arranged? To solve this problem, we can use the concept of permutations. The word "MATHS" has 5 letters, so n = 5. We need to arrange all 5 letters, so r = 5. Using the permutation formula, we get: nPr = 5! / (5 - 5)! = 5! / 0! = 5! / 1 = 5 * 4 * 3 * 2 * 1 = 120. Therefore, there are 120 different ways to arrange the letters of the word "MATHS".
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