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Please enter your name here. You have entered an incorrect email address! January 19, April 26, Load more. Commercial organizations use mathematics in accounting, inventory management, marketing, sales forecasting, and financial analysis.
Mathematics typically used in commerce includes elementary arithmetic, elementary algebra, statistics and probability. Business management can be done more effectively in some cases by use of more advanced mathematics such as calculus, matrix algebra and linear programming.
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We begin by motivating the basic counting principle which is useful in solving a wide variety of problems. Theorem 3. Solution Figure 3. By the Basic Counting Principle, the total number of routes is 2. Example 3. Solution The procedure of labeling a chair consists of two tasks, namely, assigning one of the letters and the assigning one of the possible integers.
The quiz consists of three multiple-choice questions with four choices for each. Successively answering the three questions is a three-stage procedure. Likewise, each of the other two questions can be answered in four ways.
By the Basic Counting Principle, the number of ways to answer the quiz is 4. Answering the quiz can be considered a two-stage procedure. From part a , the three multiple-choice questions can be answered in 4.
Each of the true-false questions has two choices true or false. By the BCP, the number of ways the entire quiz can be answered is This is a three-stage procedure. By the Basic Counting Principle, the total number of three-letter words is 5. For instance Example 3. Thus we have the following result: The number of permutations of n objects taken r at a time is given by n! Solution We shall consider a slate in the order of president, vice president, secretary and treasurer.
Each ordering of four members constitutes a slate, so the number of possible slates is 20! The other nine can then be arranged in 9! Figure 3. Thus, the number of distinguishable permutations is not 7! We now give a formula to enable us solve similar problems. Using this formula, the answer to Example 3. Note that the word 7! Therefore there are 3! Massasauga is a white venomous snake indigenous in North America. Therefore there are 10! Therefore there are 7! Remark 3.
In how many ways can this be done? Obviously, order in which people are placed into the rooms is of no concern. The cells remind us of those permutations with repeated objects. In how many ways can this be done.
Solution Here people are placed into three cells matatus : 6 in cell1, 5 in cell2, 4 in cell 3. Thus there are 15! Listing all such combinations and all permutations of these combinations, we obtain a list of permutations of the n objects taken r at a time. Thus n n! Solution Order is not important because no matter how the members of a committee are ar- ranged, we have the same committee. Thus, we simply have to compute the number of combinations of 20 objects taken four at a time, 20 20!
Solution Once the group of 8 has been selected then the remaining 12 children will automatically comprise the other group. For the selection of those to join the group of 8 we have two cases: i. The total number of ways will be 18 18 18! There are bananas, apples, pears, kwi, apricots, and oranges in the house. In how many ways can a selection of four pieces of fruit be chosen?
Solution Note that only the selection of varieties not which person eats what fruit is of interest here. This is a combination with repetition problem. Thus the solution is 9! How many distinguishable arrangements of the word are possible?
The number of distinguishable arrangements is 11! We must choose three men from 20 and two women from This can be done in 20 The answer is 12 20 12 Use your result to approximate 1. Note that 1. Therefore 1. Solution Using Theorem 3. Determine the values of 4 P, C, 2 C n 2. In a question examination, each question is worth 10 points and is marked right or wrong.
Considering the individual questions, in how many ways can a student score 80 or better? In a certain African country, vehicle license plates consist of two letters of the alpha- bet, followed by four digits from 0 to 9, 0 and 9 included. A carton contains 24 light bulbs, one of which is defective. In how many ways can three bulbs be selected? In how many ways can three bulbs be selected if one is defective.
A group of tourists is composed of six from Nairobi, seven from London and eight from New York. In how many ways can a committee of six tourists be formed with two people from each city?
In how many ways can a committee of seven tourists be formed with at least two tourists from each city? In how many ways can Francesca line up 12 of these books on her bookshelf?
There are 8 persons, including a married couple Mr and Mrs Bush, from which a committee of 4 has to be chosen. In how many ways can the committee be chosen a. If both Mr and Mrs Bush are excluded, b. Bush is excluded. Let n be a non-negative integer. Now replace the second nk with an equivalent expression]. Relationships between elements of sets occur in many contexts. Such rela- tionships are represented using the structure called a relation. The most direct way to express a relationship between two sets is to use ordered pairs made up of two related elements.
Example 4. This means for instance that 0Ra, 1Ra, etc. Relations can be represented graphically, using arrows to represent ordered pairs. Figure 4. Note that the terms symmetric and antisymmetric are not opposites since a relation can have both of these properties or may lack both of them.
Reasons: 1, 1 , 2, 2 , 3, 3 are not in R1. Is R an equivalence relation? Now, suppose that aRb. Hence, bRa. It follows that R is symmetric.
Hence aRc. Thus, R is transitive. Suppose that R1 consists of all ordered pairs a, b , where a is a student who has taken course b, and R2 consists of all ordered pairs a, b , where a is a student who requires course b to graduate. That is, b is an elective course that a has taken. The set A is called the domain of f and the set B is called the co-domain of f and is denoted Dom f.
The subset of B of those elements that appear as second components in the ordered pairs of f is called the range of f and is denoted by Ran f or f A. Thus, Ran f is the set of images of the elements of A under f. It is customary to use x for what is called the independent variable and y as the dependent variable. This means that y depends on x. Clearly f is onto. No negative numbers appear in Ran f. Clearly f is a bijection, since it is both one-to-one and onto.
The identity function on A is usually denoted by 1A. That is, composition of functions is not, in general, commutative. Proof a. Remark 4. Theorem 4. Solution It is easy to verify that both f and g are bijective and so each has an inverse. According to 4. We now swap the roles of x and y. We call this set the graph of f. Graphs of such functions are straight lines with gradient m and y-intercept c. The range of f is R itself. Note that the identity and constant functions are linear functions.
The graphs of these functions are parabolas. Thus a linear function is a polynomial of degree 1. A polynomial of degree 2 is a quadratic, while a cubic is a polynomial of degree 3. The graph of an even function is symmetrical about the y-axis. A function is rational if it can be expressed as q x , where p x and q x are polynomial functions.
All other functions are transcendental. Sines, cosines, tangents, etc are called trigonometric or circular functions.
Besides these functions we also have the hyperbolic functions: sinh x, cosh x, etc. Determine whether or not each of the following relations is a function. This is a relation but not a function. This is a relation from Q to Q but not a function. For each of the following functions, determine whether it is one-to-one, onto and determine its range. Determine whether or not each of the following functions is a bijection.
Solution Easy and hence left as an exercise. Find the following a. Let W, Z denote the set of whole numbers and integers, respectively. Give an example of: a. Trigonometry is based on certain ratios, called trigonometric functions, which are very useful in surveying, navigation and engineering. These functions also play an important role in the study of vibratory phenomena such as sound, light, electricity, etc.
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