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Introduction
Arrangement and selection of objects are the central ideas of this chapter on permutations and combinations. They are widely applied in solving problems of probability, genetic engineering and life sciences.
Fundamental Principle of Counting
Suppose two events E1 and E2 are to be performed in sequence, then if E1 can be performed in 'm' ways and for each of these ways E2 can be performed in 'n' ways, then the sequence E1E2 can be performed in 'mn' different ways. This is known as the Fundamental Principle of Counting.
Factorial Notation
Let n be a positive integer. The continued product of first n natural numbers is called factorial n and is denoted as n!.
Permutations
The different arrangements that can be made with a given number of things taking some or all of them at a time are called permutations.
The symbol nPr or P(n,r) is used to denote the number of permutations of n things taken r at a time.
Circular Permutations
When things are arranged in places along a line with first and last place, they form a linear permutation. So far we have dealt only with linear permutations. When things are arranged in places along a closed curve or a circle, in which any place may be regarded as the first or last place, they form a circular permutation.
Combinations
The selection of a number of things taking some or all of them at a time are called combinations.
The number of ways of selecting r things out of n dissimilar things is denoted by C(n, r) or nCr.
Summary
The fundamental principle of counting (F.P.C) states that if an operation can be performed in m different ways and if for each such choice, another operation can be performed in n different ways, then both operations, in succession can be performed in exactly mn different ways. The principle can also be generalized, for even more than two operations.
Conclusion
In this chapter, we have learnt the application of permutations and combinations, the fundamental counting principle and relation between nCr and nPr.
