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

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  \Rightarrow f(x1) < f (x2) for all xl, x2 \in (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  \Downarrow <img src="increasing decreasing function.jpg" alt="increasing decreasing function">

Read more

Continuity of functions-IB Maths topics

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  {\lim }\limits_{x \to c} f(x) = f(c) .

 

symbolically f is continuous at x = c if  {\lim }\limits_{x \to c - h} f(c + h) = {\lim }\limits_{x \to c - h} f(c - h) = f(c).

 

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.

<img src="continuous functions.png" alt="continuous functions">

Read more

IB Mathematics (part-2)-Domain and Range of a function

In IB Mathematics both HL and SL, functions are one of the most important areas because they lie at the heart of much of mathematical analysis.here I am discussing domain and range of a function

Domain of a Function

Suppose I say that f is a real function.This means that for real input, the output should be real. For example

 F(x)=√x

if F is real, then x can only take non-negative values because only then the output will be real. Set of all real values of R is called domain

“Domain is the set of all possible inputs for which the output is real ”

In some cases, x is defined explicitly. for example,

                                     y=f(x)

                                       =x²;    1<x>2 here domain is defined explicitly as (1,2)

If no domain is mentioned explicitly, the domain will be assumed to be such that “F” produces real output

These are a few examples

  1. y=f(x)=√x
    Domain D= x≥0

Read more