Fundamental Principles of Counting



  • Sets and Relations
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  • Complex Numbers 33
  • Sequences and Series
    • Concept of Sequences
    • Geometric Progression (G.P.)
    • General Term Or the nth Term of a G.P.
    • Sum of the First n Terms of a G.P.
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    • Recurring Decimals
    • Harmonic Progression (H. P.)
    • Types of Means
    • Special Series (Sigma Notation)
  • Locus and Straight Line
    • Locus
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    • General Form Of Equation Of Line
  • Determinants
    • Determinants
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    • Application of Determinants
    • Cramer’s Rule
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  • Limits
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  • Differentiation
    • The Meaning of Rate of Change
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  • Partition Values
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  • Measures of Dispersion
    • Measures of Dispersion
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    • Karl Pearson’S Coefficient of Skewness (Pearsonian Coefficient of Skewness)
    • Features of Pearsonian Coefficient
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  • Bivariate Frequency Distribution and Chi Square Statistic
    • Bivariate Frequency Distribution
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  • Correlation
    • Correlation
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    • Concept of Correlation Coefficient
    • Scatter Diagram
    • Interpretation of Value of Correlation Coefficient
  • Permutations and Combinations
    • Introduction of Permutations and Combinations
    • Fundamental Principles of Counting
    • Concept of Addition Principle
    • Concept of Multiplication Principle
    • Concept of Factorial Function
    • Permutations
    • Permutations When All Objects Are Distinct
    • Permutations When Repetitions Are Allowed
    • Permutations When All Objects Are Not Distinct
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  • Commercial Mathematics
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  • Tree Diagram 
  • Addition Principle 
  • Multiplication principle


Let us consider the following problem. Mohan has 3 pants and 2 shirts. How many different pairs of a pant and a shirt, can he dress up with? There are 3 ways in which a pant can be chosen, because there are 3 pants available. Similarly, a shirt can be chosen in 2 ways. For every choice of a pant, there are 2 choices of a shirt. Therefore, there are 3 × 2 = 6 pairs of a pant and a shirt.
Let us name the three pants as `P_1, P_2 , P_3`  and the two shirts as `S_1, S_2`. Then, these six possibilities can be illustrated in the Fig.

Let us consider another problem of the same type. 
Sabnam has 2 school bags, 3 tiffin boxes and 2 water bottles. In how many ways can she carry these items (choosing one each).
A school  bag can be chosen in 2 different ways. After a school bag is chosen, a tiffin box can be chosen in 3 different ways. Hence, there are 2 × 3 = 6 pairs of school bag and a tiffin box. For each of these pairs a water bottle can be chosen in 2 different ways. Hence, there are 6 × 2 = 12 different ways in which, Sabnam can carry these items to school. If we name the 2 school bags as `B_1, B_2`, the three tiffin boxes as `T_1, T_2, T_3` and the two water bottles as `W_1, W_2`, these possibilities can be illustrated in the Fig. 

“If an event can occur in m different ways, following which another event can occur in n different ways, then the total number of occurrence of the events in the given order is m×n.” 
In the first problem, the required number of ways of wearing a pant and a shirt was the number of different ways of the occurence of the following events in succession:
(i) the event of choosing a pant 
(ii) the event of choosing a shirt. 
In the second problem, the required number of ways was the number of different ways of the occurence of the following events in succession: 
(i) the event of choosing a school bag 
(ii) the event of choosing a tiffin box 
(iii) the event of choosing a water bottle.
Here, in both the cases, the events in each problem could occur in various possible orders. But, we have to choose any one of the possible orders and count the number of different ways of the occurence of the events in this chosen order.

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