The Arena

 The sign ! After a positive integer is a very familiar sight for the students of elementary mathematics. Factorial is defined only for positive integers, for a positive integer say n, its factorial value is represented as  n! = 1*2*3.....*(n-1)*n Factorial is the heartbeat of the counting problems i.e permutations and combinations and thats why it is so important to pay it a little more attention. We will start with some important attributes of factorials. 1. Factorial n represents the total number of ways in which any n distinct objects can be arranged in a row. it is easy to deduce , let the positions be numbered from 1 to n , then first object can be placed in n number of ways, second object can be placed in (n-1) remaining number of ways..and so on so that for the last object only one place will be left. Therefore total number of ways will be  = n*(n-1)*......*3*2*1 = n! Now again lets say we want to choose any r objects at a time from the above given group of n dist...

 Numbers and especially the whole numbers are of special significance because as a child we are first taught about them only. The most common facet of mathematics is counting and it makes sense to revisit such a fundamental concept again. So lets start with something familiar, A positive number, lets say 25. It is a perfect square. It has divisors in 1 , 5 and in 25 itself. So it is breakable number , meaning it can be broken down as a product of lower integers , i.e 25= 5*5.  Now what about the number 23, you can see that it is not divisble by any integer except 1 and 23 itself and hence it is unbreakable. Such type of numbers are called the PRIME Numbers. Prime numbers are the building blocks of the number system because every whole number can be uniquely expressed as a product of prime numbers. (This statement is known as fundamental theorem of Airthmetic) See for example 100 = 2*2*5*5 44 = 2*2*11, 99= 3*3*11 etc (Here you see the integer 2 as a prime number, indeed it is a...

  one day your sorrows will go you will be carried away by the happiest glow your heart will be pacified your agony will blow you can imagine the above moment but there is a long way to go you have to defeat all to achieve your goal but need to conquer your own fear first of all you may got hurt during your quest but it will be worthy for your taste you want to taste the success in the end so you have to go without an enervation and bend the quest may claim your fortune but it may give you the moments and opportunes the onus is on you to prove yourself worthy without that you will be ugly and earthy you can be somebody someday your peers can take pride from you in their own way you can find your love meanwhile you may laugh with her for your whole life you can see now what you can get you can feel now what you don't have so you get the reason to live and fight you have to prove yourself worthy for that right -Parag