## 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

=

In latest notation system   is also known as C(n;r) or

Properties of  – It’s a very useful and interesting Mathematical tool. It has following properties.

(i)

(ii)

(iii)      known as Pascal’s law

(iv) r.

(v)

(vi) If n is even then we should put r=n/2 for maximum value of   and if n is odd then   is greatest when r=

(vii) In the expansions of  if we put x=1 then

## 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 :

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

(iii)    you can watch this video for the explanation.

## Applications of Integration

In my previous posts, we discussed Definite and Indefinite Integrations. Now we shall learn about Applications of Derivatives. Initially, we shall discuss “Area Under Curves”.

Area Under Curve-: If we want to calculate the area between the curves y=f(x) and y=g(x) then there are actually two cases-

First Case when   Below is the figure showing this case

here area under these  two curves

The second Case When  Below figure shows this case

## Definite Integration

In the previous post, we discussed indefinite integration. Now we shall discuss definite integration

► Definite Integration- We already know that      this c here is an integral constant. we are not sure about its value. This c is the reason we call this process indefinite integration. But suppose we do our integration between certain limits like:-

here a lower limit while b higher limit

=g(b)-g(a)

You can clearly see that this function is independent of ‘c’. Means we can be sure about its value so this type of integration is called  Definite Integration.

►Definite Integration of a function f(x) is possible in [a,b] if f(x) is continuous in the given interval

►If f(x), the integrand, is not continuous for a given value of x then it doesn’t mean that g(x), the integral, is also discontinuous for that value of x.

► Definite integration of a function between given limits like             Algebraic sum of areas bounded by the given curve f(x) and given lines x=a and x=b. That’s why the answer for definite integration problems is a single number.

► If  that shows a few things:-

(i) The lines between which area is bounded are co-incident(a=b)

(ii) Area covered above the x-axis=Area covered below the x-axis that means positive part of area and negative part of area is equal

(iii) there must be at least one solution/root to f(x) between x=a and x=b(this is something we study in ROLE’S THEOREM in detail)

► If given function f(x) is not continuous at x=c then we should write

► If given function f(x) > or <0 in any given interval (a,b) then   >0 or <0 in given interval (a,b)

► If given function f(x)  g(x) in the given interval (a,b) then

in the given interval

► If we integrate the given function f(x) in the given interval (a,b) then

Some More Properties of Definite Integration:- Read more

## 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:-

If

then       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

## Increasing and Decreasing Functions

### Increasing and decreasing functions

This is my third post in the series of “Applications of derivatives”. The previous two were based on “Tangent and Normal” and “Maxima and Minima”.In this post, we shall learn about increasing and decreasing functions. That is one more application of derivatives.

Increasing and Decreasing Functions- We shall first learn about increasing functions

Increasing Function-

(a) Strictly increasing function- A function f (x) is said to be a strictly increasing function on (a, b) if x1< x2  f(x1) < f (x2) for all xl, x2(a, b).Thus, f(x) is strictly increasing on (a, b) if the values of f(x) increase with the increase in the values of x.Refer to the graph in below-given figure

## Mathematics-

In my previous post, we discussed how to find the derivative of different types of functions as well as the geometrical meaning of differentiation. Here we are discussing  Applications of Derivatives in IB Mathematics
There are many different fields for the Applications of Derivatives. We shall discuss a few of them-

Slope and Equation of tangents to a curve- If We draw a tangent to a curve y=f(x) at a given point   , then

The gradient of the curve at given point=the gradient of the tangent line  at given  point

and we already discussed that slope or gradient of the tangent at given point

m=

=()

Finally to find the equation of tangent we use the slope-point form of equation

The major part of this concept is also discussed in the previous post. We should also remember following points while solving these types of questions.

(i) If two lines are parallel to each other, their slopes are always equal
i.e
(ii) If two lines are perpendicular to each other, the product of their  slopes is always -1

(iii) If a line is passing through two points    and    then, slope of the line

# Continuity of functions-

The word continuous means without any break or gap. Continuity of functions exists when our function is without any break or gap or jump . If there is any gap in the graph, the function is said to be discontinuous.

Graph of functions like sinx,cosx, secx, 1/x etc are continuous (without any gap) while greatest integer function has a break at every point(discontinuous).

1. A function f(x) is said to be continuous at x = c,  if  .

symbolically f is continuous at x = c if .

It should be noted that continuity of a function at x = a is meaningful only if the function is defined in the immediate neighborhood of x = a, not necessarily at x = a.

# Limit of a function

Limit of a function f(x) is said to exist as,  when

finite quantity.

Fundamental Theorems On Limits :

Let    &     If l & m exists then :

(i) f (x) ± g (x) = l ± m

(ii) f(x). g(x) = l. m

(iii)   provided

(iv)    where k is a constant.

(v)   provided f is continuous at        g (x) = m

Standard Limits :

(a)  and Where x is measured in radians

(b)  both are equal to e

(c) then this will show that

(d)  and   (a finite quantity) then

where z=

(e)  where a>0. In particular

Indeterminant Forms:

etc are considered to be indeterminant values

We cannot plot  on the paper. Infinityis a symbol & not a number. It does not obey the laws of elementary algebra.

+=

×

(a/) = 0 if a is finite v is not defined

a b =0,if & only if a = 0 or b = 0  and  a & b are finite.

Expansion of function like Binomial expansion, exponential & logarithmic expansion, expansion of sinx , cosx , tanx should be remembered by heart & are given below:

(i)  ex =1+x/1!+x3/3!+x4/4!……

(ii)  ax=1+(xloga)/1!+ (xloga)2/2!+ (xloga)3/3!+ (xloga)4/4!+……….where a > 0

(iii)   ln(1-x)=x-x2/2+x3/3-x4/4……….    where -1 < x  1

(iv)  ln(1-x)=-x-x2/2-x3/3-x4/4……….     where  -1 x < 1

(v )

(vi)

(v)

In next post, I will discuss various types of limit problems, their solutions and L’ Hospital’s rule.In the meantime, you can solve these basic questions from this PDF. This PDF is for beginners only. I will post difficult and higher level questions in the next post on this topic

## 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

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

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