Permutation

Definition:

An n -element permutation of a set with n objects is simply called a permutation, denoted by . The number of permutations of a set containing n elements is

Example:

The 3 digits 1, 2, 3 can be arranged in different orders:

123, 132, 213, 231, 312, 321

Definition:

The number of permutations, of r objects chosen from n is the number of possible arrangements when r objects are to be selected from a total of n and arranged in order. This number is

Remark:

Instead of , the symbols and P (n, r) are frequently used to denote the number of permutations of n objects taken r at a time. Different authors frequently use different symbols.

Example:

Three students, Kanat, Askhat, and Marat must be scheduled for a job interviews. In how many different orders can this be done?

Solution:

The number of different orders is equal to the number of permutations of the set {Kanat, Askhat, Marat}. So there are 3!=6 possible orders for the interviews.

Example:

If 5 persons are to pose for a photograph by standing in a row, how many different arrangements are possible?

Solution:

Since we are arranging 5 “objects” 5 at a time, the number of different arrangements is given by .

Example:

If five boys and five girls sit in a row in a random order, what is the probability that no two children of the same sex sit together?

Solution:

There are 10! ways for 10 persons to sit in a row. In order that no two of the same sex sit together, boys must occupy positions 1, 3, 5, 7, 9 and girls positions 2, 4, 6, 8, 10, or vice versa. In each case there are possibilities. So, the desired possibility is .

Theorem: The number of distinguishable permutations of n objects of k different types, where are alike, are alike,…… are alike and is

.

Example:

How many different 10- letter codes can be made using three a ’s, four b ’s, and three c ’s?

Solution:

By theorem, the number of such codes is .

2.7.2. Combination

If order is of no importance, then we have a combination rather a permutation. A combination of n objects taken r at a time is a selection of r objects taken from the n, without regard to the order in which they are selected or arranged. Order is irrelevant.

Definition:

The number of combinations, , of r objects chosen from n is the number of possible selections that can be made. This number is

Remark:

Some other symbols which are used to denote the number of combinations of n objects taken r at a time are , C (n, r) and .

Example:

If a club has a membership of ten, then how many three-man committees are possible?

Solution:

Order is not important, so this is combination problem. The number of possible committees is equal to the number of ways three persons can be selected from ten persons, namely .

We have:

Example:

In how many ways can two math and three biology books be selected from eight math and six biology books?

Solution:

There are possible ways to select two math books and possible ways to select three biology books. Therefore, by counting principle, is the total number of ways in which two math and three biology books can be selected.

Example:

A box contains 24 transistors, four of which are defective. If four are sold at random, find the following probabilities:

a) Exactly two are defective b) None is defective

c) All are defective d) At least one is defective

Solution:

There are ways to sell four transistors, so the denominator in each case will be =10626

a) Two defective transistors can be selected as and two nondefective ones as . Hence

b) The number of ways to choose no defective is .

Hence

c) The number of ways to choose four defectives from four is , or 1. Hence

d) To find the probability of at least one defective transistor, find the probability that there are no defective transistors, and then subtract that probability from 1.

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