Fundamental Principles of Counting
Multiplication Principle: Suppose an operation A can be performed in m ways and associated with each way of performing of A, another operation B can be performed in n ways, then total number of performance of two operations in the given order is mxn ways. This can be extended to any finite number of operations.
Addition Principle: If an operation A can be performed in m ways and another operation S, which is independent of A, can be performed in n ways, then A and B can performed in (m + n) ways. This can be extended to any finite number of exclusive events.
The continued product of first n natural number is called factorial ‘n’.
It is denoted by n! or n! = n(n – 1)(n – 2)… 3 × 2 × 1 and 0! = 1! = 1
Each of the different arrangement which can be made by taking some or all of a number of objects is called permutation.
Permutation of n different objects
The number of arranging of n objects taking all at a time, denoted by nPn, is given by nPn = n!
The number of an arrangement of n objects taken r at a time, where 0 < r ≤ n, denoted by nPr is given by
Properties of Permutation
Important Results on Permutation
The number of permutation of n things taken r at a time, when repetition of object is allowed is nr.
The number of permutation of n objects of which p1 are of one kind, p2 are of second kind,… pk are of kth kind such that p1 + p2 + p3 + … + pk = n is
Number of permutation of n different objects taken r at a time,
When a particular object is to be included in each arrangement is r. n-1Pr-1
When a particular object is always excluded, then number of arrangements = n-1Pr.
Number of permutations of n different objects taken all at a time when m specified objects always come together is m! (n – m + 1)!.
Number of permutation of n different objects taken all at a time when m specified objects never come together is n! – m! (n – m + 1)!.
Each of the different selections made by taking some or all of a number of objects irrespective of their arrangements is called combinations. The number of selection of r objects from; the given n objects is denoted by nCr, and is given by
Properties of Combinations