combination formula

In the previous post we have discussed about Differentials Calculus and In today's session we are going to discuss about combination formula. Combination is a technique we use for the purpose of counting the number of ways we can have probable arrangements for the given events. In probability also we would find the use of combination a lot. The combination formula is written as follows:
If we have N events and we wish to select b possible events from N, then we would write it as:
^NC _b = N! / b! (N – b)!
To solve the formula for combination we must have knowledge of how to expand a factorial of any number. Let us consider an example of combination as follows:
Example: In how many ways we can select 2 black and 4 red balls from a container which contains 10 black and 10 red balls.
Solution: It is been given that the container contains 20 balls in total out of which 10 are black and the rest 10 are red. To select 2 black balls from the container we must use the combination formula for 10 black balls. So, the number of ways we can choose two black balls out of 10 is equal to ¹⁰ C ₂. Similarly, the number of ways to choose 4 red balls from the 10 red balls is equal to ¹⁰ C ₄. So, finally we get the total number of ways of selecting 2 black and 4 red balls from the container of 10 black and 10 red balls as:
¹⁰C ₂ + ¹⁰ C ₄ = 10! / 2! (8)! + 10! / 4! (6)! = 10 x 9 /2 + 10 x 9 x 8 x 7 /24 = 45 + 210 = 255 ways.
Next we discuss the concept of how to calculate density. Density can be calculated by using a simple formula: Density = Mass /Volume. These concepts are important from the perspective of icse syllabus for class 8.

Differentials Calculus

Differentials Calculus in mathematics is a part of calculus which has many important uses and can be used to derive quantities like change of rate, slope of a tangent at any point on the curve, maximum and minimum values of function etc. For instance, suppose we are given a continuous function whose slope, critical point and max or min values are to be calculated, we use application of differentiation. The first differential is used for the purpose of finding the rate change and the critical points whereas on differentiating the 1st differential again we get the max & min values.

Let us consider an example of differential calculus as follows: Suppose we have a function h (x) = 5 x⁵, then its first derivative would give us h’ (x) = 25 x⁴. Substituting h’ (x) = 0 we get x = 0. Thus the critical point is obtained at x = 0. In case the h’ (x) would have resulted in a fixed value; we say that there exist no min or max values for the function. In our example we are not getting any fixed value. Next we differentiate the 1st derivative again to get the maximum or minimum values. If the h’’ (x) comes – ve, then it is maximum value and for h’’ (x) + ve we get minimum value. In case h’’ (x) = 0 the value resembles an inflection point, and can be mistreated. In our example we would get h’’ (x) = 100 x3. Next we find out the maximum or the minimum value for the function.

Next we study how to add fractions with unlike denominators: It is a very simple concept and for solving such problems we take the LCM of the denominators. For instance: 12 /13 + 11 /10 = (120 + 143)/130 = 263/130. These concepts are important and are discussed in the icse sample papers in  detail. In the next session we will discuss about combination formula. 

Identity matrix

Matrix can be considered as an array where elements are stored in the form of rows and columns. Here elements in matrix can be considered as any type like algebraic expression, symbols or numbers. Matrix can be treated as a useful way for representing the linear transformation in a simple manner. Values in a matrix can be represented as shown below.

Suppose we have some values like 1, 2, 3, 4, 5, 6, 7, 8 and 9.

We have a matrix of 3 * 3 size then it can be represent above given values in form of matrix as:

 1  2  3

 4  5  6

 7  8  9

Now we will discuss about one of the types of matrix which is known as identity matrix. According to the definition of Identity matrix we can say that it is a kind of matrix where I A = A and A I = A. If we want to simplify the definition of identity matrix then we can say that it is square matrix which has a value 1 in diagonal position of matrix from top left to down right and all remaining position carry value zero into it. Sometimes this matrix is also known as unit matrix. How a identity matrix looks like is given below:

                             1  0  0

Identity Matrix =          0  1  0

                             0  0  1

Some properties of identity matrix are described below:

A ) When unit matrix includes the product of two square matrix then it can be said that it generates the inverse of one another.

B ) When unit matrix is multiply by itself then it generates the same output.

C ) This kind of unit matrix has a positive definite square root value.

In mathematics, the concept of Math Order of Operations defines a priority level of any operation that describes which operation should be performed first when any mathematical expression carry multiple operations.

Free download cbse books are available on various websites. In the next session we will discuss about Differentials Calculus.

Mathematical Induction

Mathematical induction is a process which can be applied to set of general statements for sets of positive integers or their connected sequences.

In this we find either a given statement is true for entire natural numbers or not. As we discussed that in the case of natural numbers we will include only positive numbers. This should not be interpreted as a form of inductive reasoning. The easiest form of induction characterizes that either a statement that includes a natural number 'n' holds for all values of 'n' is true or not.

It can be understood with help of an example:

Example: Show that sum of first 'n' odd integers is n²?

It can be written as 1 + 3 + 5 + 7 + ... + (2n - 1) = n² for all positive integers.

Let’s have a proof of above expression: First find the value of A (n). It is equals to A (n): 1 + 3 + 5 + 7 + ... + (2n - 1) = n², then in basic step we have to show A (1) is true. So it can be written as: Trivial: 1 = 1².

In Inductive step we need to show that A (n) is true for all 'n' therefore A (n + 1) is also true for all 'n'. Let A (n) is true so 1 + 3 + 5 + 7 + ... + (2n - 1) = n²,

Also write it as A (n + 1): 1 + 3 + 5 + 7 + ... + (2n - 1) + (2n + 1) = (n + 1)² follows:

 1 + 3 + 5 + 7 + ... + (2n - 1) + (2n + 1) = (n + 1)²,

It can also be written as:

n² + (2n + 1) = (n + 1)², this is the proof of mathematical induction.

Precipitation Reaction is used in formation of solid in a solution or inside another solid.

Cbse syllabus for class 11 is available online..

 

Slope

In the previous post we have discussed about Proof of bay s theorem and In today's session we are going to discuss about,Slope of a line can be defined as ratio of change in vertical axis to change in horizontal among two points on a line. If the slope of line is undefined or not defined then it is known as vertical line and if the slope of line is given as 0, then it is known as horizontal line. Formula to find the slope equation is given as:

                   y = mx + c,

Here value of ‘m’ stands for slope of line and y- intercept is given by ‘c’. It is also described by using the formula given below:

m = k₁ – k₂ / l₁ – l₂,

Here ‘m’ stands for slope of line and a₁, a₂ are the points defined on y- axis and b₁, b₂ are the points defined on x- axis. Above equation can also be written as:

     m = k₂– k₁ / l₂ – l₁,

In this gradient of a line is also defined which is denoted by the given formula:

         m = tan θ,

Let’s take an example. It will be clear with help of an example:

Example 1: - Calculate slope of the line segment that join the points (4, -7) and (-7, 4)?

Solution: - As we see above that it is a line segment that join the points (4, -7) and (-7, 4). As we know that formula to find the slope is given as:

m = k₂ – k₁ / l₂ – l₁, here value of k₁ = 4, l₁ = -7 and k₂ = -7, l₂ = 4

Now put given values in formula to find its value. On putting value in formula to get result.

m = -7 – 4 / 4 – (-7), on further solving we get:

m = - 11 / 11,

So here we get the value of 'm' is -1. In this way we can find out the value of 'm'.

Mann Whitney Test can be used to see either two independent samples of observations are drawn from same distributions.

To prepare for 10 th board exams focus on cbse sample papers for class 10.

Proof of bay s theorem

In mathematics, we will study different theorem. Here we will understand the concept of bay s theorem. Bay’s theorem can have two distinct interpretations. It is an important concept of Bayesian statistics and has different properties in the field of science and engineering. Formula that is used to solve the probability, that name was given after the 18 th – century by the great scientist British mathematician Thomas bayes. Now we talk about the formula used in Bay’s theorem which is given below :
K (I / J) = K (I ∩ J) / K (J) = K (I) * K (J | I) / K (J)
Now we will understand the proof of bay’s theorem: Suppose we have X and Y_j be two sets. Then the conditional probability requires that:
P (X ∩ Y_j) = P (X) P (Y_j | P), here the symbol ‘∩’ represented as intersection (‘and’) and also said :
P (X ∩ Y_j) = P (Y_j ∩ X) = P (Y_j) P (X | Y_j), therefore it can be written as:
P (Y_j ∩ X) = P (Y_j) P (X | Y_j) / P (X), now assume Z = ∪_{i = 1}ⁿX_i, so X_i is an event in Z and X_i ∩ X_j = ⱷ for i ≠ j, then we can write it as:
X = X ∩ Z = X ∩ (∪_{i = 1}ⁿX_i) = ∪_{i = 1}ⁿ(X ∩ X_i),
P (X) = P (∪_{i = 1}ⁿ(X ∩ X_i)) = ∑_{i = 1}^N P (X ∩ X_i), it can also be written as:
P (X) = ∑_{i = 1}^N P (X_i) P (X | X_i),
P (X_i | X) = P (X_i) P (X | X_i) / ∑_{i = 1}^N P (X_i) P (X | X_i) this is the proof of bay’s theorem.
Paper Chromatography is technique that is used for separating and identifying mixtures that can be colored, especially pigments. Before entering in the board exam please prefer cbse sample papers. It is helpful for examination point of view.

definition of a function

In the previous post we have discussed about How to Find the Domain of a Function and In today's session we are going to discuss about definition of a function. In this blog we will discuss the definition of a function. Function is used to show relationship among set of inputs and set of outputs in which every value of input is related to exactly one value of output. In other words a relation defined from I to J such that a sub set of I * J is called as function from I to J. Let's us understand the concept of function and its types. There are different types of function which are given as:

One – one function

Many one function

Onto function

Now we will have small introduction about all its types.

One – one function (it is also called as injection function): - A function f: I → J is said to be one – one function if every input value of element 'I' has different image in 'J'. So it can be written as:

f : I → J is one – one if value of 'i' not equal to 'j'. (i ≠ j) → f (i) ≠ f (j) for all ij Ԑ I.

Many one function: - A function f :I → J is said to be many one function if two or more elements of set 'I' have same images in 'J'. In mathematical form it can be written as:

f: I → J is a many one function if there exist a, b Ԑ I such that a ≠ b but f (a) = f (b).

Onto function: - A function f: I → J is said to be onto function or it is also said to be 'surjection' if every value of element 'J' is image of some element of 'I' that is if f (I) = J, and range of 'f' is co – domain of function 'f' or in other words elements of 'J' has no pre – image in element 'I'. This is all about types of functions.

We will study Primary Structure of a Protein in chemistry. Primary structure is join together by covalent or peptide bonds. cbse sample paper for class x is important for class 10 th student.