Permutations refers to the different ways in which a number of a number of objects can be arranged in a different order
Example: Suppose there are two things x and y, they can be arranged in to two different ways i.e,. xy and yx . These two arrangements is called permutation
Similarly x, y and z
xyz, xzy, yxz, yzx, zxy, zyx is 6 arrange permutation
(if we want to have two things only from x,y,z then xy,xz,yz,yx,zx,yz only in this case)
July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
Formulae 2: Finding the number of permutations of ‘n’ things taken ‘r’ at a time, given that each of the elements can be repeated once, twice….up to ‘r’ times
Or
‘ n’ things taken all at a time of which ‘p’ are alike, ‘q’ others are alike and ‘r’ others alike
Example 1: How many permutations are possible of the letters of the word PROBABILITY when taken all at a time?
Combinations refers to the number of arrangements which can be made from a group of things irrespective of the order
Combinations differ from permutations in that one combination such as xyz may be stated in the form of several permutations just by rearranging the orders as : xyz, xzy, yxz, yzx, zxy, zyx
Note: All of these are one combination but they are six permutations
IMP Note: The number of permutations is always greater than the number of combinations in any given situation since a combination of n different things can be generate n factorial permutations
Let C ( n , r ) be the number of ways to generate unordered combinations
The number of ordered combinations (i.e. r -permutations) is P ( n , r )
The number of ways to order a single one of those r -permutations P ( r,r )
The total number of unordered combinations is the total number of ordered combinations (i.e. r -permutations) divided by the number of ways to order each combination
Thus, C ( n,r ) = P ( n,r )/ P ( r,r )
July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
The number C(n , r) can be obtained by constructing a triangular array.
The row 0, i.e., the first row of the triangle, contains the single entry 1 . The row 1, i.e., the second row, contains a pair of entries each equal to 1 .
Calculate the n t h row of the triangle from the preceding row by the following rules:
The technique known as divide and conquer can be used to compute C(n , r ).
In the divide-and-conquer technique, a problem is divided into a fixed number, say k , of smaller problems of the same kind.
Typically, k = 2 . Each of the smaller problems is then divided into k smaller problems of the same kind, and so on, until the smaller problem is reduced to a case in which the solution is easily obtained.
The solutions of the smaller problems are then put together to obtain the solution of the original problem.
July 14, 2010 Basic Quantitative Techniques - RVMReddy - ABS
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