Permutations and Combinations-algebra tutors

Permutations and Combinations(part-2)

In my previous post, we discussed the fundamental principle of counting and various methods of permutations. In this post, I shall discuss combinations in details.

Meaning of Combination- If we are given a set of objects and we want to select a few objects out of this set, then we can do it by many different ways. These ways are known as combinations.
Example- If we are given three balls marked as B, W and R and we want to select two balls then we can select like this- BW, BR, WR.

These are known as the combination of this selection.

Combination of n different objects taken r at a time when repetition is not allowed– If repetition is not allowed the number of ways of selecting r objects out of a group of n objects is called  {}^n{c_r}

{}^n{c_r}=   \frac{{n!}}{{r!\left( {n - r} \right)!}}

In latest notation system  {}^n{c_r} is also known as C(n;r) or   \left( \begin{array}{l} n\\ \\ r \end{array} \right)

Properties of  \left( \begin{array}{l} n\\ \\ r \end{array} \right)– It’s a very useful and interesting Mathematical tool. It has following properties.

(i) {}^n{c_r}{}^n{c_{n - r}}

(ii)  {}^n{c_n} = {}^n{c_0} = 1

(iii)  {}^n{c_r} + {}^n{c_{r - 1}} = {}^{n + 1}{c_r}    known as Pascal’s law

(iv) r. {}^n{c_r} = n{}^{n - 1}{c_{r - 1}}

(v)  \frac{{{}^n{c_r}}}{{{}^n{c_{r - 1}}}} = \frac{{n - r + 1}}{r}

(vi) If n is even then we should put r=n/2 for maximum value of  {}^n{c_r} and if n is odd then  {}^n{c_r} is greatest when r= \frac{{{n^2} - 1}}{4}

(vii) In the expansions of {(1 + x)^n} if we put x=1 then

{}^n{c_0} + {}^n{c_1} + {}^n{c_2} + ....... + {}^n{c_n} = {2^n}

{}^n{c_0} + {}^n{c_2} + {}^n{c_4} + ......... = {2^{n - 1}}

{}^n{c_1} + {}^n{c_3} + {}^n{c_5} + ......... = {2^{n - 1}} Read more

Permutation and Combination

Permutations and Combinations-

‘Permutations and Combinations’ is the next post of my series Topics in IB Mathematics.It is very useful and interesting as a topic. It’s also very useful in solving problems of Probability. To understand Permutations and Combinations, we first need to understand Factorial.

Definition of Factorial-  If we multiply n consecutive natural numbers together, then the product is called factorial of n. Its shown by n! or by

for example :       n! = n(n - 1)(n - 2)(n - 3)..........3.2.1

Some Properties of Factorials-
(i) Factorials can only be calculated for positive integers at this level. We use gamma functions to define non-integer factorial that’s not required at this level
(ii) Factorial of a number can be written as a product of that number with the factorial of its predecessor    n! = n[(n - 1)(n - 2)(n - 3)..........3.2.1]

 = n(n - 1)!

(iii)  0! = 1  you can watch this video for the explanation.

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Indefinite Integration-Topics in IB Mathematics

Indefinite Integration

After a long series on differentiation and ‘Application of derivatives‘, we shall now discuss Indefinite Integration. It consists of two different words indefinite and integration.
First of all, we shall learn about Integration.

 Integration is the reverse process of differentiation so we can also call it as antiderivative. There is one more name for it, that is Primitive.
If f & g are functions of x such that g'(x) = f(x) then the function g is called a Primitive Or Antiderivative Or Integral of  f(x) w.r.t. x and is written symbolically as:-

\int {f\left( x \right){\rm{ }}dx = g\left( x \right) + c}

If    \frac{d}{{dx}}\left\{ {f(x) + c} \right\} = f'(x)

then  \int {f'\left( x \right){\rm{ }}dx = f\left( x \right) + c}      here c is just an arbitrary constant. Value of c is not definite that’s why we call it Indefinite Integration.

Techniques  Of  Integration-: There are a few important techniques used to solve problems based on integration

(i) Substitution or  Change of Independent Variable- If the derivative of a function is given in the question, then we should use the method of substitution to integrate that question. Read more

How to solve trigonometric problems based on complimentary anngles?(concept-3)

IB Maths tutors give great importance to Trigonometry.

Trigonometry is one of the fascinating branches of Mathematics. It deals with the relationships among the sides and angles of a triangle.Word trigonometry was originated from the Greek word, where, ‘TRI‘ means Three‘GON‘ means sides and the ‘METRON’ means to measure. It’s an ancient and probably most widely used branch Mathematics. For basic learning, IB Maths Tutors divide trigonometry in two part:-

1. Trigonometry based on right triangles

2. Trigonometry based on non-right triangles.

Here, we are discussing trigonometry based on non-right triangles only.

In the third article of this series, we will discuss problems based on complementary angles

In the third article of this series, we will discuss problems based on complementary angles

<img src="right triangle.jpg" alt="right triangle">

In this right triangle Sin A=BC/AC & Cos C=BC/AC   clearly: Sin A=Cos C  In the given triangle A+C=90° so we can write C=(90°-A). This gives us freedom to write Sin A=Cos (90°-A) similarly we can write these relationships     Read more

How to solve basic problems in trigonometry?(concept-2)

Mathematics tutors give great importance to Trigonometry

Trigonometry is one of the fascinating branches of Mathematics. It deals with the relationships among the sides and angles of a triangle.Word trigonometry was originated from the Greek word, where, ‘TRI‘ means Three‘GON‘ means sides and the ‘METRON’ means to measure. It’s an ancient and probably most widely used branch Mathematics. For basic learning, I am dividing trigonometry in two part:-

1 Trigonometry based on right triangles

1 Trigonometry based on non-right triangles.

In this post, we will discuss problems based on trigonometric ratios of a few specific angles like 0°,30°, 45°, 60° and 90°. Mathematics tutors use different tricks to form this table. I will discuss my tricks in a separate post

<img src="trigonometric table.jpg" alt="trigonometric table"> Read more

How to solve basic problems in trigonometry?(concept-1)

Mathematics tutors give great importance to Trigonometry

Trigonometry is one of the fascinating branches of Mathematics. It deals with the relationships among the sides and angles of a triangle.Word trigonometry was originated from the Greek word, where, ‘TRI‘ means Three‘GON‘ means sides and the ‘METRON’ means to measure. It’s an ancient and probably most widely used branch Mathematics. For basic learning, I am dividing trigonometry in two part:-

1 Trigonometry based on right triangles

1 Trigonometry based on non-right triangles.

In this post, I will only discuss Trigonometry based on right triangles.

In a right triangle, there are three sides hypotenuse (the longest side), adjacent side(base) and the opposite side(perpendicular).

                             <img src="right triangle.png" alt="right triangle"> Read more

Ib Maths Tutors-Types of functions(part-3)

Types of Functions-

IB Maths Tutors should give twenty-two hours for teaching functions and equations as per IBO recommendations. This is my third article on functions in the series of ib mathematics

IB Maths Tutors should give twenty hours in teaching functions and equations.This is my third article on functions in the series of ib mathematics

For the sake of a comprehensive discussion, some standard functions and their graphs are discussed here

For the sake of a comprehensive discussion, some standard functions and their graphs are discussed here.

1.Greatest Integer Function–  This is an interesting function. It is defined as the largest     integer less than or equal to x

                                                         y = [x].

For all real numbers, x, this function gives the largest integer less
than or equal to x.

For example:   [1] = 1      [2.5] = 2      [4.7] = 4      [5.3] = 5
Beware!    [-2] = -2      [-2.6] = -3      [-4.1] = -5      [-6.5] = -7

domain=R
range=Z

<img src="ibmathstutors.jpg" alt="ibmathstutors">

greatest integer function

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IB Mathematics Tutors- types of mathematical function(part-1)

IB Mathematics Tutors should give a fair number of hours in teaching functions.This is my third article on functions in the series of ib mathematics

For the sake of a comprehensive discussion, some standard functions and their graphs are discussed here

For the sake of a comprehensive discussion, some standard functions and their graphs are discussed here
1.Constant Function-
                                                F(x)=k
Domain= ℝ    

                                            Range    = {k}                                       

<img src="constant-function.jpg" alt="constant-function">

constant-function

                                   
Range    = {k}

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