Fundamental Principles of Counting
Fundamental Principles of Multiplication: If there are two jobs such that one of them can be completed in m ways and when t has been completed in any one of these m ways, second job can be completed in n ways, then the two jobs in succession can be m x n ways.
If there are n jobs J₁, J₂, … such that job Ji can be performed independently in mi ways i = 1, 2, 3, … n. then the total number of ways in which all the jobs can be performed is m₁ x m₂ x, … x mn.
Example: In a class there are 10 boys and 8 girls. The teacher wants to select a boy and a girl to represent the class in a function. In how many ways can the teacher make this selection?
Solution: Here, the teacher is to perform two jobs
- Selecting a boy among 10 boys
- Selecting a boy among 8 girls
The first of these can be performed in 10 ways and the second one in 8 ways. Therefore, by the fundamental principle of multiplication, the required number of ways is 10 x 8 = 80.
Fundamental Principle of Addition: If there are two jobs such that they can be performed independently in m and n ways respectively then either of the two jobs can be performed in (m + n) ways.
Example: In a class there are 10 boys and 8 girls. The teacher wants to select either a boy or a girl to represent the class in a function. In how many ways can the teacher make this selection?
Solution: Here the teacher is to perform either of the following jobs.
- Selecting a boy among the 10 boys
- Selecting a girl among the 8 girls
The first of these can be performed in 10 ways and the second in 8 ways. Therefore, the fundamental principle of addition either of the two jobs can be performed in (10 + 8) = 18 ways. Hence, the teacher can make the selection of either a boy or a girl in 18 ways.