# The Fundamental Principle Of Association

The lemma stated below is known as the fundamental principle of association.

Lemma:

Suppose if one operation can be performed in m ways and after it has been performed in any of these ways, a second operation can be performed in n ways, then the total number of ways of performing the two operations will be m x n.

Example:

Three families went out for a tour and arrived in a hotel. There are 5 vacant rooms in the hotel. In how many ways can the families take up their quarters, each in different room?

Solution:

Clearly, the first family can choose any one of the 5 different rooms; hence this family has 5 different selections and when it has made its selection in any one way, the second family will have a choice of 4. Then, the first two families can select their rooms in 5 x 4 ways. Again, with any one choice of the first two, the third family can select their rooms in 3 ways. Hence, the three families can select their rooms in 5 x 4 x 3 = 60 ways.

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