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Fundamental Principles of Counting

1. Fundamental Principle of Multiplication: If there are two operations that can be performed independently in m and n ways respectively then the two operations in succession can be performed in (m × n) ways.

Example:  In a room, there are four entrance doors and three exit doors. In how many ways can a person enter the room and then come out?

Solution:  Clearly, a person can enter the room through any of the four entrance doors. So, there are 4 ways of entering the room.

After entering the room, the person can come out through any of the three exit doors. So, there are 3 ways of coming out.

Hence, by the fundamental principle of multiplication, the number of ways in which a person can enter the room and then come out = (4 × 3) = 12.

2. Fundamental Principle of Addition: If there are two operations that can be performed independently in m and n ways respectively then either of the two operations can be performed in (m + n) ways.

Example: In a class there are 16 boys and 9 girls. The teacher wants to select either a boy or a girl as a class representative. In how many ways can the teacher make the selection?

Solution:   Here the teacher is to perform either of the following two operations:

(i) selecting 1 boy out of 16 boys, and

(ii) selecting 1 girl out of 9 girls.

The first of these can be performed in 16 ways and the second in 9 ways.

By the fundamental principle of addition, either of the two operations can be performed in (16 + 9) ways = 25 ways.

Hence, the teacher can make the selection of 1 boy or 1 girl in 25 ways.

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