Permutation

In probability theory, the arrangement of items into different ways is very important. There are two methods, permutation and combination.

Mathematically if the order (arrangement) does not matter, it is called combination and if the order (arrangement) does matter it is called permutation.

A permutation (act of permuting) is an ordered arrangement (rearrangement) of a set of items or objects. Or we can define it as

Possible ways in which a finite number of items can be arranged in different sequences with the consideration of the order of their occurrence are called permutation. Consider the three letters a, b and c. one possible permutation of these letters is abc, another permutation is acb. All the different permutations of these letters are:

Abc        acb         bca         bac         cab         cba

These three objects (letters) are arranged in six possible different way (3!=6).

Rule 1:

The number of permutations of n different items taken n at a time can be denoted by nPn, where

nPn = n(n – 1) (n – 2) . . . 2.1

The formula shows that in selecting the first of the n items, there are n choices. Once the first item is selected, there remains n – 1 choices for the second item or n (n – 1) possible choices for the first two items. For third item there are n – 2 possible choice. It continues until all the item are selected in this fashion. We can write nPn in factorial notation as

nPn = n!

Rule 2:

The number of permutations of n different items taken r at a time is denoted by nPr, where

\[^n P_r= n(n-1)(n-2)\cdots (n-r+1)\]

Or

\[^nP_r=\frac{n!}{(n-r)!}\]

This formula is similar to previous one as once  r – 1 items have been selected, the number of different choices for the rth item equals n – (r – 1) or n – r +1.

Example:

A candidate for presidential election would like to visit seven cities prior to the next primary election data. But, for him, it will be possible to visit only three of the cities. How many different itineraries (routes, schedules) can he and his staff consider?

Here order of visit is important in planning a itinerary, therefore the number of possible itineraries is equal to the number of permutations of seven cities taken three at a time i.e

\begin{align*}

^nP_r = \frac{n!}{(n-r)!}\\

^7P_3 &= \frac{7!}{(7-3)!}\\

&=\frac{7.6.5.4.3.2.1}{4.3.2.1}\\

&=7.6.5 =210

\end{align*}

 

 

Equation: Solving First Degree Equation

An equation is equality of two algebraic expression stated in terms of two or more variable in it. Any equation (linear) that can be written or rearranged in the form of ax+b=0 , is called first degree equation. The following are the examples of equation having equality sign between two expressions:

i)             4x – 25 = 20 – 5x                                                one variable (x) first degree equation

ii)            $\frac{2x – 5y + 8z}{5}=250$                        three variable (x, y, z) first degree equation

iii)           $x^2 – 2y = – 10                                                 two variable (x, y) and second degree equation

The solution of any equation consists on set of numbers (values) makes the equation true when the values of variable are substituted in the equation. The values satisfying or making the equation true are referred to as the roots of the equation. In other words roots are the values of the variable(s) which satisfy the equation.

In equation (i) putting x=5, makes the equation true, while putting any other value such as x=0 in equation (i) results in 15= 20, which is not true, so x=0 is not the root of the equation, while x=5 is the root of the equation.

Note that we can have three types of equations.

(1)                 Identity equation
An identity is an equation which is true for all values of the variable, for example

\[8x + 12 = \frac{16x + 6}{2}\]

Another example is 5(x + y) = 5x + 5y

(2)                  Conditional Equation
A conditional equation is true for only a limited number of values of the variable, for example

\[x + 3 = 5\]

This equation is true only when x has value of 2 i.e. x=2.

(3)                False Statement or contradiction

A false or contradiction statement is always false or it never true, that is there are no values of the variables which makes the both side of the equation equal, for example

\[x = x + 5\]

This equation indicate that the both sides of equation will not be equal.

Solving any equation is called the process of finding the roots or solution set of an equation, if any root(s) exist. Usually, to solve an equation, we manipulate or rearrange the equation.

Rules for solving first degree equation

  • You can add the same number on both sides of the equation.
  • You can subtract the same number on both sides side of the equation.
  • You can multiply with the same number on both sides (except 0) of the equation
  • You can divide with the same number on both sides (except 0) of the equation

Origin Of Percentages

Percent or percentage comes from Latin phrase per centum, is a number or ratio expressed as a fraction of 100 and is used to communicate information to other people in every day life. It is usually denoted by percent sign “%”, and its simplest meaning are per hundred, e.g. 43% (read as forty three percent i.e. 43 out of 100).

In market, research surveys and reports, news channels etc., we usually see numbers in percentage form such as in market different business owner announce sale discount in form of percentage for example 15% discount on a particular product in off season. Similarly many of the jobs or commission are based on percentage sales of the product. Banks gave interest in percentage on amount deposited in particular type of account. We are usually comfortable when discount, sales tax, interest, commission etc. are in percentages i.e. they are easy to understand and interpret because of in percent notation.

The percent or percentage values can be computed by multiplying ratio by 100.

Example 1: Calculate 20% of 80

25\% &= \frac{25}{100}\\
&=\frac{25}{100}\times 80=20

 

Example 2: 15% of 250 educational toy were defected. How many of toy were defective in number?

15\% &= \frac{15}{100}\\
&=\frac{15}{100}\times 250=37.5

Compuing Percentage of Percentage

To compute percentage of a percentage, simply convert both of the percentage into fractions of 100 and then multiply them for example

Example 3: what is 50% of 30%?

\[\frac{50}{100} \times \frac{30}{100}=0.5*0.3 = 0.15 = \frac{0.15}{100}=15%\]

Percentage Increase or Decrease

Problems involving change either increaser or decrease are common in especially business applications. In case of an increase, the amount of change is added to the original quantity, while in case of decrease, the amount of change is subtracted from the original quantity.

Example 4: The assets of a business increased from $120,000 to $340,000. What is the rate (percent) of change?

Final quantity &= 340,000\\

Amount of change &= 340,000 – 120,000 = 220000\\

Rate of change &= \frac{amount of change}{Original quantity}\\

&=\frac{220,000}{120,000}\times=183.33%

Example 5: After a discount of 25%, the sale price of a suitcase is $200. What was the original price of the suitcase?

Let original price is $x$
price after discount is $200

Price after discount price = original price – percentage of discount

\[200 = x – 0.25x = 0.75x\]

\[X = 200/0.75 = 266.667\]

Inequalities

An inequality is a conditional equation in which two quantities are not equal as inequality are expressed using symbols of <, >, ≥, ≤ and ≠. We can also say that an inequality shows a relation that holds two different quantities. Some examples of inequality are

i)                    3 < 5                      3 is less than 5

ii)                  X > 100                  the value of X is greater than 100

iii)                0 < y < 10              The value of y is greater than 0 and less than 10 (or we can say that the value of y lies between 0 and 10)

Strict Inequality

Inequalities are called strict inequality if quantities being compared can never be equal to each other. In above example, all inequalities are strict inequality.

Conditional Inequality

A conditional inequality is true under certain condition or constraint. For example, inequality in example (ii) is conditional inequality, because when x is greater than 100, inequality becomes true.

Double Inequality

Double inequality are of the form a < y < b, meaning that all real numbers lies between a and b. Example (iii) is double inequality.

Not strict Inequality

Another type of inequality is expressed by relation using symbols ≤ and ≥. These kind of inequality relationships allows the certain possibilities that two quantities are equal, for example x + 3 ≥ 5.

Solving Inequality

Inequalities can be solved in similar fashion as equation are solved. Solving inequalities means that we are attempting to find the set of values that must satisfy the inequality. We use same algebraic operations to solve inequality as we use in solve equations. The only difference is that when working with inequality, the sign or direction of inequality reverse when inequality is multiplied or divide by negative number. For example, the sign of the inequality -2 < 3 changes if we multiply it by (-1), i.e. 2 > -3.

All inequalities can be shown on number line to represent the set of values satisfying inequality on graph or number line. Some examples for double inequality are:

Inequalities

 

Teaching Math using data related to natural disasters

Now days Disaster data is being used in the Asia Pacific Economic Cooperation (APEC) region to teach math to elementary and high school students. Mathematicians, Educators and scientist are trying to make students get involved with math and also training teachers to improve math skills. Learning real world math will help the learners to use math skills in managing natural disasters events, better preparedness for future uncertain natural disasters.

In Japan in 2011 after the earthquake and tsunami and immense flood in Thailand before that, realize the experts that scientific and statistical data about the disaster can be used to educate children about disaster awareness, survival and math.

APEC funded specialists from different region of Japan, Russia, United States, Thailand, Indonesia, Malaysia, Vietnam, Singapore,  Philippines, South Korea, Hong Kong, Chile and Peru to work together in developing math textbooks, for learning and teaching mathematics skill having different sort of mathematical activities related to type and severeness of natural disaster. Customized proposed curriculum according to the country’s need are developed, helping teacher to plan their class lessons in the light of the recommendations by specialist and try it out in a classroom setting.

Instead of teaching children about math, they should also taught students to learn about how to survive in natural disaster situation by applying mathematical techniques. Real-life problem and problems related to children’s own family and friends will help them to apply math in different real life situations such as planning and scheduling traffic, and city planning etc.

Exposing children to disaster data and showing them how mathematics and science can be used in different real life situation is necessary to survive and be aware about the disaster. One must start learning about hazards through mathematics, and mathematics through examples of hazards at a young age to be prepared for future.

For school going students, basic and comprehensive mathematical skill such as addition, subtraction, multiplication, division, unit conversion, decimal and percentage, algebraic expression, equation, logic and problem solving etc. should be taught with different applied mathematics examples related to disaster. For advance students high-level mathematical techniques, functional relationship between variables (such as linear, quadratic, exponential functions), calculus, multivariate techniques,  stochastic programming and modeling techniques along with characteristics of different natural disasters should be taught.

Classroom Activities and Lesson Plans

  • Example about how mathematics can be used to describe nature, and different natural disasters
  • Examples about measurement unit used
  • Examples about function and mathematical model
  • Classifying and identifying relationship between variables related to natural disasters
  • Creating and interpreting graphs