## Business Mathematics and Statistics

**Contents:**

About the Author ii

Preface vii

Syllabus for Honours & General ix

Question Pattern xi

Chapter 1: Permutations and Combinations 1.1–1.60

1.1 Introduction 1.2

1.2 Meaning of Permutation and Combination 1.2

1.3 Differences Between Permutation and Combination 1.3

1.4 Identification of Permutation and Combination 1.3

1.5 Factorials 1.3

1.6 Fundamental Principles of Counting 1.4

1.6.1 Multiplication Principle 1.4

1.6.2 Addition Principle 1.4

1.7 Different Rules on Permutation 1.5

1.7.1 Permutations of Different Things 1.5

1.7.2 Permutation of Things Not All Different 1.6

1.7.3 Repeated Permutations 1.7

1.7.4 Circular Permutations 1.8

1.7.5 Restricted Circular Permutations 1.8

1.7.6 Restricted Permutations 1.10

1.8 Different Rules on Combination 1.11

1.8.1 Combinations of Different Things 1.11

1.8.2 Restricted Combinations 1.13

1.8.3 Total Number of Combinations of Different Things 1.14

1.8.4 Total Number of Combinations of Things

not all Different 1.15

1.8.5 Division of Groups 1.15

Illustrative Examples 1.17

Permutations 1.17

Combinations 1.28

Exercise 1.40

A. Short Type 1.40

B. Short Essay Type 1.44

C. Advanced Problems 1.51

D. Multiple Choice Questions (MCQs) 1.52

Chapter 2: Set Theory 2.1–2.58

2.1 Introduction 2.2

2.2 Definition of a Set 2.2

2.3 Elements of a Set 2.3

2.4 Notation of a Set 2.3

2.5 Representation of a Set 2.3

2.5.1 Roster or Tabular Method 2.3

2.5.2 Rule or Set-builder Method 2.4

2.6 Types of Sets 2.5

2.6.1 Finite Set 2.5

2.6.2 Infinite Set 2.5

2.6.3 Singleton Set 2.5

2.6.4 Null Set or Empty Set 2.5

2.6.5 Equal Sets 2.6

2.6.6 Equivalent Sets 2.6

2.6.7 Subset 2.6

2.6.8 Power Set 2.7

2.6.9 Universal Set 2.8

2.7 Venn Diagram 2.8

2.8 Overlapping Sets 2.9

2.9 Disjoint Sets 2.9

2.10 Cardinal Number of a Set 2.9

2.11 Operations on Sets 2.10

2.11.1 Union of Sets 2.10

2.11.2 Intersection of Sets 2.10

2.11.3 Difference of Sets 2.11

2.11.4 Symmetric Difference of Two Sets 2.12

2.11.5 Complement of a Set 2.13

2.12 Laws of Algebra of Sets 2.13

2.13 Verification of Associative, Distributive and De Morgan’s

Laws Using Venn Diagram 2.14

2.14 Proof of Associative, Distributive and De Morgan’s Laws

from Definition 2.18

2.15 Cartesian Product of Sets 2.21

2.16 Some Important Results on Number of Elements in Sets 2.23

Illustrative Examples 2.23

Exercise 2.44

A. Theoretical 2.44

B. Short Type 2.44

C. Short Essay Type 2.48

D. Practical 2.50

E. Advanced Problems 2.52

mF. Multiple Choice Questions (MCQs) 2.53

Chapter 3: Binomial Theorem 3.1–3.39

3.1 Introduction 3.1

3.2 Binomial Theorem for a Positive Integral Index 3.2

3.2.1 Statement of the Theorem 3.2

3.3 Properties of Binomial Expansion (x + a)n 3.3

3.4 Applications of the Binomial Theorem 3.3

3.5 Some Particular Cases 3.4

3.6 Properties of the Binomial Coefficients 3.5

3.7 General Term 3.5

3.8 Middle Term 3.7

3.9 The ‘rth’ Term From the End 3.8

3.10 Equidistant Terms 3.8

Illustrative Examples 3.9

Exercise 3.24

A. Short Type 3.24

B. Short Essay Type / Problem Type 3.25

C. Advanced Problems 3.31

D. Multiple Choice Questions (MCQs) 3.33

Chapter 4: Logarithm 4.1–4.37

4.1 Introduction 4.1

4.2 Definition 4.2

4.3 Properties of Logarithms 4.3

4.4 Types of Logarithms 4.3

4.5 General Laws of Logarithm 4.4

4.6 Characteristic and Mantissa 4.6

4.7 Determination of Characteristic 4.6

4.8 Determination of Mantissa 4.7

4.9 Calculation of Logarithm of a Number 4.8

4.10 Determination of Mantissa of a Number Consisting of

More than Four Digits 4.8

4.11 Antilogarithm 4.10

Illustrative Examples 4.12

Exercise 4.27

A. Short Type 4.27

B. Short Essay Type/Problem Type 4.28

C. Advanced Problem 4.31

D. Multiple Choice Questions (MCQs) 4.32

Chapter 5: Compound Interest and Annuities 5.1–5.75

Compound Interest

5.1 Introduction 5.2

5.2 Present Value of a Single Amount 5.7

5.3 Nominal Rate and Effective Rate of Interest 5.8

Annuities

5.4 Introduction 5.8

5.5 Types of Annuities 5.9

5.6 Immediate or Ordinary Annuity 5.10

5.6.1 Amount or Future Value of Immediate or

Ordinary Annuity 5.10

5.6.2 Present Value of Immediate or Ordinary Annuity 5.10

5.7 Deferred Annuity 5.12

5.7.1 Amount of Deferred Annuity 5.12

5.7.2 Present Value of Deferred Annuity 5.12

5.8 Annuity Due 5.13

5.8.1 Amount of an Annuity Due 5.13

5.8.2 Present Value of an Annuity Due 5.14

5.9 Present Value of Perpetual Annuity or Perpetuity 5.14

5.10 Continuous Compounding 5.15

5.10.1 Derivation of Continuous Compounding Formula 5.15

5.11 Valuation of Simple Loan 5.16

5.12 Valuation of Debenture/Bond 5.19

5.12.1 Present Value of a Redeemable Bond or Debenture 5.19

5.12.2 Present Value of a Perpetual or Irredeemable Bond

or Debenture 5.20

5.13 Sinking Fund 5.21

Illustrative Examples 5.22

Compound Interest 5.22

Annuities 5.36

Exercise 5.51

Compound Interest 5.51

A. Short Type 5.51

B. Short Essay Type/Problem Type 5.53

C. Advanced Problems 5.56

Annuities 5.57

A. Short Essay Type/Problem Type 5.57

B. Miscellaneous Problems 5.64

C. Multiple Choice Questions (MCQs) 5.65

Chapter 6: Correlation and Association 6.1–6.66

6.1 **Bivariate Data** 6.2

6.2 Correlation 6.2

6.2.1 Definition of Correlation 6.2

6.2.2 Types of Correlation 6.2

6.3 Co-efficient of Correlation 6.4

6.4 Methods of Computing Co-efficient of Correlation 6.5

6.4.1 Scatter Diagram 6.5

6.4.2 Pearson’s Product Moment Co-efficient of Correlation 6.9

6.4.3 Spearman’s Rank Correlation Co-efficient 6.19

6.5 Association of Attributes 6.24

6.5.1 Introduction 6.24

6.5.2 Classification and Notations 6.25

6.5.3 Order of Classes and Class Frequencies 6.25

6.5.4 Relationship between the Class Frequencies 6.26

6.6 Consistency of Data 6.26

6.7 Association 6.27

6.7.1 Difference between Association and Correlation 6.27

6.8 Methods of Studying Association 6.27

6.8.1 Comparison of Observed and Expected

Frequencies Method 6.27

6.8.2 Proportion Method 6.29

6.8.3 Yule’s Coefficient of Association 6.30

Illustrative Examples 6.32

Exercise 6.46

A. Theory 6.46

B. Practical/Short Type 6.47

C. Short Essay/Problem Type 6.49

D. Multiple Choice Questions (MCQs) 6.58

Chapter 7: Regression Analysis 7.1–7.41

7.1 Regression 7.1

7.2 Regression Analysis 7.2

7.3 Types of Regression Techniques 7.2

7.3.1 Linear Regression 7.2

7.3.2 Least Squares Method 7.2

7.3.3 Non-linear Regression 7.3

7.4 Regression Line 7.3

7.5 Derivation of the Regression Equations 7.5

7.6 Regression Coefficients 7.7

7.7 Properties of Regression Lines (Including Correlation

Coefficient and Regression Coefficients) 7.8

7.8 Identification of Regression Lines 7.10

7.9 Uses of Regression 7.13

7.10 Difference Between Correlation and Regression 7.13

7.11 Explanation of Having Two Regression Lines 7.14

Illustrative Examples 7.14

Exercise 7.28

A. Theory 7.28

B. Short Type 7.28

C. Short Essay / Problem Type 7.30

D. Miscellaneous 7.35

E. Multiple Choice Questions (MCQs) 7.37

Chapter 8: Index Numbers 8.1–8.75

8.1 Introduction 8.2

8.1.1 Meaning of **Index Number** 8.3

8.2 Types of Index Numbers 8.3

8.2.1 Price Index Numbers 8.3

8.2.2 Quantity Index Numbers 8.3

8.2.3 Value Index Numbers 8.3

8.3 Characteristics of Index Numbers 8.4

8.4 Uses of Index Numbers 8.4

8.5 Problems of Constructing Index Numbers 8.5

8.6 Common Notations 8.6

8.7 Methods of Constructing Index Numbers 8.6

8.7.1 Aggregative Method 8.7

8.7.2 Relative Method 8.11

8.8 Quantity Index Numbers 8.14

8.9 Tests of Adequacy of Index Number 8.16

8.9.1 Time Reversal Test 8.16

8.9.2 Factor Reversal Test 8.16

8.9.3 Circular Test 8.17

8.10 Errors in Index Numbers 8.20

8.10.1 Formula Error 8.20

8.10.2 Homogeneity Error 8.20

8.10.3 Sampling Error 8.21

8.11 Chain Base Index Numbers 8.21

8.12 Base Shifting, Splicing and Deflating 8.23

8.13 Consumer Price Index or Cost of Living Index 8.28

8.13.1 Construction of Consumer Price Index Numbers 8.28

8.13.2 Methods of Construction 8.29

8.13.3 Applications and Uses 8.30

Illustrative Examples 8.31

Exercise 8.54

A. Theory 8.54

B. Short Type 8.56

C. Short Essay Type 8.57

D. Multiple Choice Questions (MCQs) 8.68

Chapter 9: Time Series Analysis 9.1–9.58

9.1 Introduction 9.2

9.2 Time Series 9.2

9.3 Time Series Analysis 9.3

9.4 Objectives of Time Series Analysis 9.3

9.5 Utility of Time Series Analysis 9.3

9.6 Components of Time Series 9.4

9.7 Time series Models 9.6

9.7.1 Additive Models 9.6

9.7.2 Multiplicative Model 9.6

9.8 Measurement of Trend 9.7

9.8.1 Semi-Average Method 9.7

9.8.2 Moving Average Method 9.9

9.8.3 Method of Least Squares 9.14

9.8.4 Measurement of Seasonal Variations 9.21

9.9 Business Forecasting 9.27

9.9.1 Methods of Business Forecasting 9.28

9.10 Steps of Business Forecasting 9.29

9.11 Limitations of Business Forecasting 9.30

Illustrative Examples 9.30

Exercise 9.43

A. Theoretical 9.43

B. Practical 9.44

C. Multiple Choice Questions (MCQs) 9.51

Chapter 10: Probability Theory 10.1–10.60

10.1 Introduction 10.1

10.2 Meaning of Probability 10.2

10.3 Importance of Probability 10.3

10.4 Some Important Terms used in Probability Theory 10.3

10.5 Different Definitions of Probability 10.6

10.5.1 Classical Definition 10.6

10.5.2 Frequency Definition 10.7

10.5.3 Axiomatic Definition 10.9

10.6 Probability of Event ‘not E’ 10.10

10.7 Odds in Favour of and Odds Against an Event 10.10

10.8 Theorem of Total Probability 10.10

10.9 Conditional Probability 10.13

10.10 Compound Probability 10.14

10.11 Probability of Independent Events 10.15

10.12 Compound Independent Events 10.16

10.13 Bayes’ Theorem 10.17

Illustrative Examples 10.19

Exercise 10.42

A. Short type 10.44

B. Short Essay Type 10.46

C. Multiple Choice Questions (MCQs) 10.52

Appendix A: Solution of C.U. Question Paper–2017

(New Syllabus) A.1–A.26

For Honours Candidates A.1

For All Other Categories of Candidates A.14

Appendix B: Log Tables B.1–B.4