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  The formula to obtain Binomial Coefficient can be written as: n! / r! × ( n - r )! , which is equal to the number of combinations. Here, n  -> number of unique objects  r  ->  number of objects that are                           considered ( r <= n  ) ____________________ Suppose n = 4 Objects that are being considered: ( 1 ) A ( 2 ) B ( 3 ) C ( 4 ) D _____________________ Let us restrict ourselves to deal with only three objects at a time. We will attempt to form combinations taking only three objects at a time. So, here r = 3 _____________________ From the ongoing discussion, n = 4 and r = 3 So, No. of combinations =                 4! / ( 3! × ( 4 - 3 )! ) = 4 Let us observe the four combinations that are formed: ________________________...

Suppose we have two Jobs A and B. A and B carry data of size 1 Kilo Byte each. Data carried by Jobs A and B has to be processed.  Let us further suppose that there are two slots ( Slot I and Slot II ) available. The important condition is that processing of Jobs A and B can occur, only if they appear in Slot I. When these jobs appear in Slot II, they are not processed.  ____________ Case I: ____________ The permutations of Jobs A and B are as: _________________             Slot I        Slot II Perm I         A         B Perm II        B           A               ________________ //Permutation- A Great Fun// //Titbits Under the Sun// ____________________ Clearly job A would be in Slot I for once.  Job B too would be in Slot I for once. Let us suppose the  data proc...

Factorial of a number is the product of that number and all the numbers below it.  Factorial 4 is symbolically expressed as: 4! To calculate Factorial 4: 4! = 4 × 3 × 2 × 1 = 24 So, in calculation of finding the Factorial of '4'  we have taken into account numbers from '4' to '1'. We have moved in descending order upto '1'. Factorial of a number can also be calculated using recursion. Recursion is a method to solve a problem. Some mathematical problems are nicely solved using recursion. Let n be a non-negative counting number. n can be zero, let us say. The factorial of n is:  n ! = n × ( n - 1 )!  By definition,  0! = 1 Let us find the factorial of ' 2' using recursion. By the formula, 2! = 2 × ( 2 - 1)! =  2 × 1 !  ------ ( i ) 1 ! = 1 × ( 1 - 1 )!= 1 × 0 !   ------ ( ii ) 0! = 1 (By definition of factorial) ----- (iii) Now, upon substitution of ( iii ) in  ( ii ), we get: 1 ! = 1 ×1 = 1 --------- ( iv ) Now, substituting ( iv ) in ( i ...

  A sum is simply calculated. Arithmetic calculation is performed in determining the sum. Calculation of sum is part of elementary mathematics. Addition is performed for finding the sum. When Sum is to be found, we must have a minimum of two numbers. When two numbers are added, we get the Sum of two numbers. As for example, 4 + 5 = 9 '4' is the augend. '5' is the addend. '9' is the Sum. The arithmetic operator '+' is used.  We can have a Sum of more than two numbers. There is no upper limit on the count of operands in calculation of Sum. The minimum count of numbers to be added is two.  In the addition of numbers, we calculate the Sum of parts. The total Sum cannot be more than the values of the individual parts. Similarly, the total Sum cannot be less than values of individual parts. So, we can say that the  Sum concept proves the truthfulness and durability of numbers. The Sum tells about the magnitude of individual parts. In this way we can say that t...

Numbers are used to express quantity. They also tell the order of things. Numbers are also used as labels. Numbers are expressed by symbols. 1, 2, 3, 4 is a representation of numbers. Quantity represented by numbers can be modified. When a quantity is modified its representation also changes. The modified quantity is represented by a different number.  Quantity changes when some addition is made, or the quantity has been lessened.  When addition is made, we can apply the arithmetic operator '+'. The method of using the operator is: Previous Quantity + Current Amount = Modified Quantity If   Previous Quantity = 2   Current Amount = 7        then                2 + 7 = 9 ( Modified Quantity ) In the above case the value of Modified Quantity will always be greater than Previous Quantity. Similarly we can use the arithmetic operator '-'.  Previous ...

Application of Power can make a thing rise. Indeed, the use of power makes sense. It gives simplicity in our attempt to represent numbers. It leads to achievement of mathematical solutions very easily and quickly. ______________________________ Note: Power is also called exponent. ______________________________ A number is given a power when it is needed. It can be said that the number has been raised to a numeric value. Any number that is given power as '0' is equal to '1'. In Mathematics '2' to the power '4' means that the number '2' must be multiplied by itself '4' times. This is written as 2^4 , or 2 × 2 × 2 × 2, which equals 16. With a power a number becomes more powerful. Power is a type of qualification for a number. The task of writing numbers has simplified and become easy with the Power concept.  Large numbers are difficult to write. Another point is that such numbers are difficult to read. The  Power concept is really very help...

Mathematical Induction is a method of supporting a mathematical statement. The method is based on producing evidence for the purpose of verification of the statement. The presented evidence serves to verify the mathematical statement. The correctness of the mathematical statement is tested by the verification. The mathematical statement is verified. The mathematical statement stands true if use of scientific verification justifies the statement. At first, the mathematical statement is verified by using a limited form of evidence. Then we go onward for further  verification. Onward verification also must justify the meaningfulness of the mathematical statement. __________________________ Principle of Mathematical Induction: Let P(n) be a statement involving the natural number n such that  P(1) is true and P(k+1) is true, whenever P(k) is true then P(n) is true for all n € N. _____________________________   First the element produced as evidence is the natural number '1'. '...