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The Fundamental Principle of Counting

published on January 13th, 2008 . by Vanaja
If an event can happen in exactly m ways, and if following it, a second event can happen in exactly n ways, then the two events in succession can happen in exactly mn ways.

Illustration.

Suppose there are five routs from A to B and three routs from B to C. In how many ways a person can go from A to C?

Since there are five different routs from A to b, the person can go the first part of his journey in 5 different ways. Having completed in any one of the 5 different ways , he has 3 different ways to complete the second part of the journey fro B to C. Thus each way of going from A to B give rise to 3 different ways of going from B to C.

There fore the total number of ways of completing the whole journey = number of ways for the first part x number of ways for the second part.
= 5 x 3=15.

Generalisation

If an event can occur in m different ways, a second event in n different ways, a third event in exactly p different ways and so on, then the total number of ways in which all events can occur in succession is mnp….

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