This document discusses the topic of combinatorics. It defines combinatorics as a branch of mathematics concerned with counting problems, arrangements, and derangements. The two main rules in combinatorics are the sum rule, which is used when events cannot occur simultaneously, and the product rule, which is used when events are independent. Examples of applying the sum and product rules to problems involving selecting students and choosing outfits are provided. The document also covers permutations, combinations, the binomial theorem, and binomial coefficients.

1. Subject :- Discrete Structure
Submitted To :-Khoubo Patidar
Name of Presenter :-Irfan khan
PATEL COLLEGE OF SCIENCE & TECHNOLOGY
INDORE

2. COMBINATORICS

3. WHAT IS COMBINATORICS ?
• Combinatorics is a branch of discrete
mathematcs which is concerned with the
counting problems, arrangement,
dearangement, etc.
• Counting is the technique are important in
computer science and mathematics.
• Two main rule in combinatorics:-
a) Sum rule
b) Product rule

4. SUM RULE
 If two events are mutually exclusive, that is
they cannot be done at the same time ,we
must apply sum rule.
 m is the number of ways to do task 1.
 n is the numer of ways to do task 2.
 Task 1 and task 2 are not simultaneously.
 s=s1Us2
|s|=|s1|+|s2|

5. PRODUCT RULE
 If two events are mutually exclusive(that is
we do them seprately ),then we apply the
product rule
 Let us consider two task :
 m is the number of way to do task 1.
 n is the number of way to do task 2.
 If task 1 and task 2 are independent.
| A×B|=|A|.|B|

6. EXAMPLE OF SUM RULE
 In a class there are 26 girls and 15 boys ,
find the number of ways of students one
students as a class moniter.
Solution :- By sum rule , the number of ways
of selecting one students (either boy and
either girls ) = 26+15
= 41

7. EXAMPLE OF PRODUCT RULE
 If one task but to do it, suppose Raju has 4
shirts and 5 jens the then how many way
wearing a dress but Raju which cloth select
to wearing a dress.
 4 Shirts &
 5 jens
 Shirts 1 Shirts 2

 |T|=|T1|×|T2|
 = 4×5
 = 20
Jens1
Jens2
Jens3
Jens4
Jens5
Jens1
Jens2
Jens3
Jens4
Jens5

8. PERMUTATION
 A Permutation of a set of object is an orderd
arrengement of the elements .
 For example, written as tuples, there are six
permutations of the set {1,2,3}, namely:
(1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and
(3,2,1). These are all the possible orderings
of this three element set
 It follow that :-
 Permutation = nPr = n! / ( n - r )!

9. EXAMPLE OF PERMUTATION
How many 4 digit no. are possible with the digit
2,34,5,6,8,9.
1.Digit can not be repeated 2.Digit can be repeated
1.Digit can not be repeated:-
7 × 6 × 5 × 4 = 840
2.Digit can be repeated :-
7 × 7 × 7 × 7 = 2401

10. COMBINATION
 Combination is set of object of an unordered
arrangement or unordered selection of
objects
 The number of way choosing elements from
S:
S={1,2,3}
=1,2 1,3 2,3
 It follow that :-
Combination = nCr =

11. EXAMPLE
 From group 12 students ,5 are to be chosen
from a competetion.
 n=12 total no. of students
 r=5 chosen the students
 =12×11×10×9×8 = 792
 5×4×3×2×1

12C5

12. BINOMIAL THEOREM
 Binomial Theorem on the expansion of
exponents or powers on a binomial expression.
This theorem was given by newton where he
explains the expansion of (x + y)n for different
values of n.
 Binomial Theorem
 For (a + b)0 = 1
 For (a + b)1 = a + b
 For (a + b)2 = a2 + 2ab + b2
 For (a + b)3 = a3 + 3a2b + 3ab2 + b3
 For (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

13. BINOMIAL THEROEM
 General formula :-
(x + y)n = nC0xn + nC1xn-1y + nC2xn-2y2 + … +
nCrxn-ryr + … + nCnxn-nyn.
Note :- There are (n+1) terms.
(a + b)3 = a3 + 3a2b + 3ab2 + b3

14. WHAT IS THE BINOMIAL COEFFICIENT?
 In (a + b)2 = 1a2 + 2ab + 1b2
1, 2 and 1 are called as the binomial coefficients of a2, ab
and b2 respectively.
 Similarly, in (a + b)3 = 1a3 + 3a2b + 3ab2 + 1b3
1, 3, 3 and 1 are called as the binomial coefficients of a3, a2b,
ab2 and b3 respectively.
 So, In general, in the expansion of (x + y)n
(x + y)n = nC0xn + nC1xn-1y + nC2xn-2y2 + … + nCrxn-ryr + … +
nCnxn-nyn.
all the terms have a constant multiplied with the variables in
the form of nCr. These nCr are called as the binomial
coefficients of different terms