Explain Counting Technique with Permutation and Combination

types of counting techniques

In this topic of Counting Techniques, we will read and compare permutations and combinations on counting and types of counting and discuss their formulas and numerical.

Define Counting Technique in statistic?

In some cases, the sample or event space is so large that it isn’t feasible to write it out. Mathematical tools can be useful to count the size of the sample space and event space in that case. Such tools are called counting techniques.

Types of Counting Techniques in Mathematical Method

There are two types of counting techniques.

  • Permutation Technique
  • Combination Technique

What is Meant by the Permutation Technique?


In the mathematical method, the number of ways to arrange things is called a permutation.

For Example:

Arranging people, digits number of anything, letters, and color are examples of permutation.

What is called Combination?


In the mathematical method, the number of ways to choose things is called a Combination.

For Example:

An example of a combination is any cricket team.

For example, if a team coach has 16 team members, then he selects 11 out of these 16 members. This process of choosing is called a combination.

Concept Of Permutation and Combination

The simple concept of permutation and combination is given in the form of graphs below.

difference between Permutation and Combination

Cases of Permutaion

There are three types of cases of permutation.

  • Repetition is allowed
  • Repetition is not allowed
  • Distinct Permutation (when they are not all difference)

Define Repetition is Allowed

Repetition is allowed: when a single word, or a group of words, is repeated for effect.

Formula for Repetition is Allowed

for example:

when tossing two coins. and the result is

HH, HT, TH, TT

examples of Repetition is allowed

Define Repetition is not allowed

Repetition is not allowed: when a single word, or a group of words, is not repeated for effect.

Formula for Repetition is not allowed

Formula for Repetition is not allowed

For Examples

when tossing two coins. and the result is

HH, HT, TH, TT

examples of Repetition is not allowed

Define Rest and Motion with Examples in Physics Basic Concept

rest and motion example

Define Rest in Physics with Examples?

definition of rest in physics with Examples?
Define Rest in Physics: When a body doesn't change its position with respect to its surrounding, it means that the body is in Rest. Surrounding means the locations in the neighborhood of that frame in which a few are extra (diverse) items are present.

Define Motion and Explain with Example?

 On the other aspect a frame or Body is called in Motion, if it modifications its role with recognize to its surrounding. It method that the body isn't always in relaxation as it adjustments its position.

What are the five Examples of Motion in Physics?

If a body does now not exchange its function, it is said to be on relaxation, and if it adjustments its role it’s far called movement. The examples of movement encompass: running, biking, jumping, swimming, consuming, ingesting, gambling, writing, typing, shifting motors, throwing balls.

Explain Rest and Motion With Example in Physics

define rest in physics

Someone sitting in a moving vehicle is at rest due to the fact she or he is not changing his or her position with admiration to different men and women who are also sitting in the car. But to an observer outdoor the car, the man or woman’s insides of the automobile are in movement.

Define Rest and Motion with Examples in Physics and Types of Motion

define rest in physics

A few objects circulate alongside a straight line, a few circulate in some different methods.. In step with this concept, there are three types of motion which might be:

  • Translatory movement (linear , random and circular)
  • Rotatory motion
  • Vibratory motion

What is meant by Translatory Motion?

In simple words:
In translational motion, a frame or body movements along a line. objects move without any rotation. This line could be straight. It can also be curved.

Which of these is an example of Translatory Motion?

The motion of a Ferris wheel is a common instance of translator motion. A strolling girl, a stone falling immediately, and a moving vehicle bus, etc..

What are the three types of Translatory Motion?

  • ⇒linear motion
  • ⇒random motion
  • ⇒circular motion

Define Linear motion in Physics?

Whilst an object moves in an instant direction, its motion is stated to be linear movement.

Examples of linear Motion

The movement of an airplane is a pleasant example of linear movement because of its actions in a straight direction.

What is Linear Motion in General form

In nature, a body could be said to be in motion if it adjustments its position with appreciation to a reference factor and time. Even as describing the linear motion we require the best one coordinate axis at the side of time to explain the motion of a particle then it is stated to be in a linear movement or rectilinear movement. In linear motion, the particles will pass from one point to some other factor both in an instant line or a curved route. Relying upon the direction of movement linear movement is further subdivided divided as:

  • Rectilinear Movement – it’s miles a route of motion in a directly line.
  • Curvilinear Motion – it’s far a path of movement in a curved line.

Define Random motion in Physics?

While a body 0r any object moves in an abnormal manner, its motion is stated to be a random movement. It way that the body’s movement is disordered.

Examples of Random Motion

The motion of molecules of oxygen, movement of birds flying in the sky. Its simple instance could be determined in our environment i. E., the motion of a kite inside the sky and movement of clouds, and many others.

What is Circular Motion? and Example

When a frame(body) or any item actions in a round path, its motion is said to be round. In simple phrases, the motion or movement of an item in a round direction is said to be around the motion of that body of the object.

Examples of circular motion

A toy education transferring on a circular track is the maximum commonplace example and planets revolving across the primary source of heat and mild (solar).

Rotatory motion Definition with example

In simple words:
The spinning motion of a body or an item around its axis is stated to be the rotatory motion of that frame or item. Axis means a particular line round which a body or object rotates.

Example of Rotatory motion

It is a very common example is the movement of a wheel approximately its axis and that of a steerage wheel.

Vibratory motion Definition with example

In simple words:
To and for a motion of an object or any frame body approximately its mean function is stated to be vibratory motion.

Example of vibratory motion

The motion of a toddler at the same time as sitting on a swing. The motion of the pendulum of a wall clock is likewise an essential example of vibratory motion. Its most simple example from our daily life is the vibration of a cellular smartphone.

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Define Matrix Order with Examples? and Types of Matrix Order in Math

What is Matrix Order?

History of Matrix Order

The idea of matrices was introduced by a famous mathematician Arther Cayley in 1857. Matrices are widely used in both physical and social science. In this article we will introduce a new mathematical form, called a matrix, that will enable us to represent a number of different qualities as a single unit.

Details of Matrix:

Before going to know about the Matrix order we can briefly understand matrices. A matrix is a rectangular array of any number or function. Matrices are 2-dimensional .it is represented by rows and columns. Rows are denoted by m and columns are denoted by n. Now we move towards matrix order. The term matrix order is defined as multiply the number of rows and columns in that matrix. We write the order of the matrix in such a manner that the number of rows first show  and then the number of columns.

Types of Matrix order in Maths

What is Matrix order in math?

A matrix is a square or a rectangular array of numbers written within square brackets or parentheses in a definite order, in rows and columns. The term matrix order is defined as the number of rows and columns in that matrix. for example

Generally, the matrices (plural of the matrix) are denoted by capital letters… A, B, C………….etc. while the elements of a matrix are denoted by small letters a, b, c…………. and numbers 1, 2, 3,……..

Easy Matrix Order Example:        

A=\begin{bmatrix}3&5\\5&1\end{bmatrix} ,

B=\begin{bmatrix}2&3&2\\2&2&1\\4&3&4\end{bmatrix} ,

C=\begin{bmatrix}1\\2\end{bmatrix} ,

D=\begin{bmatrix}1&2\end{bmatrix}

We write the order of matrix in such a manner that number of row first show  and then the number of columns

Example

Let suppose a matrix A has m number of row and n number of the column then the order of the matrix is m*n

Let A be matrix

A=\begin{bmatrix}1&2&2\\1&2&0\end{bmatrix}

Here are 2 rows and 3 columns So order of matrix A is 2*3.

Row and column matrix examples

Types of Matrix in Maths

Some important types of matrix order are as follow:

Define Row matrix with Examples

The matrix having only one row is called a row matrix.

Examples:

Example No 1.

A=\begin{bmatrix}1&2&3\end{bmatrix}

The order of this matrix is 1×3

Example No 2.

B=\begin{bmatrix}1&2&3&4&5\end{bmatrix}

.Order of this matrix is 1×5 

Define Column matrix with Examples

The matrix having only one column is called a column matrix.

Examples:

Example No 1.

A=\begin{bmatrix}1\\2\\3\end{bmatrix}

Order of this matrix is 3×1

Example No 2.

B=\begin{bmatrix}1\\2\\3\\4\end{bmatrix}

Order of this matrix is 4×1

Define Null Matrix with Examples

A matrix in which all entries are zero is called a null matrix.

Examples:

Example No 1.

A=\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix}

Order of this matrix is 3×3

Example No 2.

B=\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}

Order of this matrix is 2×3

Explain Square matrix with Examples

A matrix in which a number of rows and number of columns are equal is called a square matrix.

Examples:

Example No 1.

A=\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}

Order of this matrix is 3×3

Example No 2.

B=\begin{bmatrix}1&2\\3&4\end{bmatrix}

Order of this matrix is 2×2

Definition of Diagonal matrix with Easy Examples

A square matrix in which diagonals entries are non-zero and off-diagonals entries are zero is called a diagonals matrix

Examples:

Example No 1.

A=\begin{bmatrix}1&0&0\\0&2&0\\0&0&2\end{bmatrix}

Order of this matrix is 3×3

Example no 2.

B=\begin{bmatrix}1&0\\0&2\end{bmatrix}

Order of this matrix is 2×2

Define Upper triangular matrix with Examples in Math

A square matrix in which all entries below the main diagonals are zero is called upper triangular matrix.

Examples:

Example No 1.

A=\begin{bmatrix}1&2&2\\0&2&2\\0&0&5\end{bmatrix}

order of this matrix is 3-by-3

Example No 2.

B=\begin{bmatrix}1&1\\0&1\end{bmatrix}

Order of this matrix is 2-by-2

The upper triangular matrix must be a square matrix.

Lower triangular matrix with Examples

A square matrix in which all items overhead the main diagonals are zero is called a lower triangular matrix.

Examples:

Example No 1.

A=\begin{bmatrix}1&0&0\\1&2&0\\2&2&2\end{bmatrix}

Order of this matrix is 3-by-3

B=\begin{bmatrix}1&0&0&\\4&6&0&0\\4&4&5&0\\2&5&4&1\end{bmatrix}

The order of this matrix is 4-by- 4

A lower triangular matrix is also must be a square matrix.

What is Symmetric matrix in Math?

A square matrix in which its transpose is equal to itself is called a symmetric matrix

You can learn more about Matrix in Math on Our Website…

How we Define Hyperbolic Functions and their Simple Formulas

Hyperbolic Functions

Define Hyperbolic Function in Simple Way

  • Sin Hyperbolic Function
  • Cos Hyperbolic Function

Simply Define sinh x hyperbolic function

Sinh x = \frac12\left(e^x-\;e^{-x}\right) is called hyperbolic sine function. Its domain and range are the set of all real numbers.

Define sinh x hyperbolic function

Formula for Cosine hyperbolic function in Math with Detail

Cosh x = \frac12\left(e^x+\;e^{-x}\right) is called hyperbolic cosine function. its domain is the set of all real numbers and the range is the set of all numbers in the interval.

[1, +∞)

Formula for Cosine hyperbolic function

Four Hyperbolic Functions and Formulas and Their Details

The remaining four hyperbolic functions are defined in the terms of the hyperbolic sine and the hyperbolic cosine function as follows:

Four Hyperbolic Function Formulas are

Functionsis equal toFormulas
tanh x\frac{\sin h\;x}{\cos h\;x}\frac{e^x\;-\;e^{-x}}{e^x\;+\;e^{-x}}
sech x\frac1{\cos h\;x}\frac2{e^x\;+\;e^{-x}}
coth x\frac{\cos h\;x}{\sin h\;x}\frac{e^x\;+\;e^{-x}}{e^x\;+\;e^{-x}}
csch x\frac1{\sin h\;x}\frac2{e^x\;-\;e^{-x}}

The hyperbolic functions have the same properties that resemble to those of trigonometric functions.

Speed Velocity and Acceleration Problems in Physics

Speed Velocity and Acceleration Problems in Physics

Speed:

Objects travel a certain distance in a given time frame and this is called their speed.

Formula:

define speed and their formula in physics

Define Velocity:

Basically, velocity is just how fast a person’s displacement changes.

Formula:

best formula for velocity in physics

which explains the information needed to calculate speed and velocity in Physics?

Distance covered and time taken are required to calculate speed.

it is denoted by v and its formula is:

v\;=\;\frac st

Where S is Distance and t is time in Formula

It is necessary to take time into consideration when calculating the velocity of the moving body.

it is also denoted by v and its formula is:
v\;=\;\frac dt

when is the average velocity of an object equal to the instantaneous velocity?

The average velocity of an object is equal to its instantaneous velocity if its acceleration is zero. There cannot be any change in speed or direction of an object if its acceleration is zero. It follows that if an object is moving in the same direction or at zero acceleration, its speed or direction can’t change.

which pair of sentences is describing the same velocity?

Vehicles moving in opposite directions have the same velocity.

Essentially, acceleration is the rate at which velocity changes, while velocity is the rate at which position changes. Thus, velocity, acceleration, and position are connected.

Speed Velocity and Acceleration problems

Speed Velocity and Acceleration problems

The main difference between speed and velocity involves

Speed is defined as:
The distance covered by an object in a unit of time is said to be its speed.

speed formula:

define speed and its short formula

However, Velocity is defined as:
The rate of change of displacement of anybody is said to be velocity.

formula for velocity in physics

which explains the information needed to calculate speed and velocity?

For calculation of speed, we require distance covered and taken time

it is denoted by v and its formula is:

v\;=\;\frac st

S is Distance and t is time.

However, For the calculation of velocity, we require displacement of that moving body and take time.

it is also denoted by v and its formula is:
v\;=\;\frac dt

when is the average velocity of an object equal to the instantaneous velocity?

If the acceleration of an object is zero then the average Velocity is equal to instantaneous velocity. And if the acceleration of an object is zero there can be no change in speed or direction of that object.

which pair of sentences is describing the same velocity?

A moving car and a truck have the same velocity.

Actually, acceleration is a rate of change of velocity and velocity is a rate of change of position. So the position, velocity, and acceleration are related to each other.

Define Vertical and Horizontal Asymptotes in Calculus

Explain Vertical and Horizontal Asymptotes

What is an Asymptote in Calculus

In mathematical terms, an asymptote is a curve that approaches infinity, asymptotes are lines that appear parallel to it.

What is an Asymptote in Calculus
Asymptote in Math

Types of Asymptotes in Calculus

There are three types of Asymptote in math.

1. Horizontal Asymptote
2. Vertical Asymptote
3. Oblique Asymptote

Difference between Horizontal Vertical and Oblique Asymptotes

Difference between Horizontal Vertical and Oblique Asymptotes

How to find Horizontal Asymptote of Exponential Function?

How to find Horizontal Asymptote of Exponential Function?
It is a Horizontal Asymptote

Horizontal Asymptote: A horizontal asymptote occurs when A curve approaches some constant value b as x approaches infinity (or *infinity)..

How to find Vertical Asymptote of Exponential Function

How to find Vertical Asymptote of Exponential Function
It is a Vertical Asymptote

Vertical Asymptote: A horizontal asymptote occurs when a function of X approaches some constant, c (from the left or right) then the curve continues to infinity (or −infinity).

What is Oblique Asymptote?

What is Oblique Asymptote?
Oblique Asymptote

Oblique Asymptote: A Oblique Asymptote occur when, as x goes to infinity (or −infinity) the curve then becomes a line y=mx+b

Asymptote for a Curve Definition in Math

Definition: A straight line l is called an asymptote for a curve C if the distance between l and C approaches zero as the distance moved along l (from some fixed point on l ) tends to infinity.

The curve C can approach asymptote l as one moves along l in one direction, or in the opposite direction, or in both directions.

How to Find Equation of Horizontal Asymptote?

Suppose the equation y\;=\;f(x) of C is such that y is real and y\rightarrow a as X\rightarrow\infty or X\rightarrow-\infty then y\;=\;a is a Horizontal asymptote. For, the distance between the curve and the straight line y\;=\;a is y\;-\;a and this approaches zero as X\rightarrow\infty or X\rightarrow-\infty.

How to Find Equation of Vertical Asymptote?

If the equation of C is such that y is real and Y\rightarrow\infty or Y\rightarrow-\infty as x\rightarrow a from one side then the straight line x = a is a vertical asymptote.

To see this, observe that (1) x – a is the distance between the curve and the straight line and that this distance is supposed to approach zero (2) Y\rightarrow\infty or Y\rightarrow-\infty as x\rightarrow a , so that

\lim_{y\rightarrow\pm\infty}\;(x-a)\;=\;\lim_{x\rightarrow a}\;(x-a)\;=\;0

Thus, to locate vertical asymptotes we have to find a number a such that \lim_{x\rightarrow a}\;y\;=\;\infty\;or\;-\infty

Similarly, if y\rightarrow mx\;+\;c as x\rightarrow\infty or x\rightarrow-\infty then y\;=\;mx\;+\;c is an asymptotes ( which is neither vertical nor horizontal).

Thus, we inquire for \lim_{x\rightarrow\pm\infty}\;y is studying asymptotes.

Find Horizontal and Vertical asymptote for Algebraic equations

For Algebraic equations, we can find horizontal and vertical asymptotes as follows:

For horizontal asymptotes we write the given equation in the form x\;=\;\frac{\psi(y)}{\theta(y)} and consider those values of y for which \theta(y)\;=\;0.

Similarly, to find a verticle asyptote, we write the given equation in the form y\;=\frac{f(x)}{g(x)} and consider those value of x for which g(x) = 0.

Working Rule for Asymptotes Parallel to the Axes

In an equation of a curve, the coefficient of the highest power of x (respectively of y) equated to zero gives asymptotes (if any) parallel to the x-axis (respectively y-axis)

Homogeneous Function of Degree Example

Homogeneous Function of Degree Example

Define Homogeneous Function with Example

A function f defined by

u\;=\;f\;(x,\;y,\;z,\;…)

of any number of variables is said to be homogeneous of degree n in these variables if multiplication of these variables by any number t(\neq0) result in the multiplication of the function by t^n, i.e.,

f\;(tx,\;ty,\;tz,\;…) = t^n f\;(x,\;y,\;z,\;…) (1)

provided that (tx,\;ty,\;tz,\;…) is in the domain of f.

Taking t=\frac1x,(x\neq0) the equation (1) becomes

  • f\left(1,\;\;\frac yx,\;\frac z{x\;}\right)\;=\;\frac1{x^n}f(x,\;y,\;z,\;…)\;
  • \;f(x,\;y,\;z,\;…)\;\;=x^n\;f\left(1,\;\;\frac yx,\;\frac z{x\;}\right)\;
  • =x^ng\left(\frac yx,\;\frac zx\;…\right)

Method to Draw the Graphs of Algebraic Functions

Defined Graphs of Algebraic Functions

If f is a real-valued function of a real numbers, then the graph of f in the xy-plane is defined to be the graph of equation y = f(x). The graph of a function f is the set of point { (x, y)| y = f(x), x is in thedomain of f } in the cartesian plan for which (x, y) is an ordered pair of f.

The graph provides a visual technique for determining whether the set of points represents a function or not. if a vertical line intersects a graph in more than one point, it is not the graph of a function.

Explain is given in the figure.

Sketch the Graph of a function

A function Graph

Defined Graphs of Algebraic Functions
A Function

Is Also a Function Graph

Sketch the Graph of a function

The Sketch is not a function

The Sketch is not a function
not a function

Best Method to Draw the graph

To draw the graph of y = f(x), we give arbitrary value of our choice to x and find the corresponding values of y. In this way we get ordered pairs \left(x_1\;,\;y_1\right) , \left(x_2\;,\;y_2\right) , \left(x_3\;,\;y_3\right) etc. These orders pairs represent points of the graph in the cartesian. We have these points and join them together to get the graph of the function.