Consider this. You have four units A, B, C and D. You are asked to select two out of these four units. How do you go about this particular task? Will your methodology remain the same, if you are asked that you should select two units, but they should be according to some predefined criteria? Definitely, it differs. In this part we look at two techniques called Permutations and Combinations, which help us solve problems like these.
Before we start looking at permutations and combinations, let us acquaint ourselves with an important principle. It says: if an operation (first) has been performed in say 'm' ways and a second operation which can be performed in 'n' ways, then both the operations can be performed in m x n ways. The explanation is as follows.
The first operation can be performed in any one of the given m ways. After performing this operation in any one of the m ways, the second operation can be performed in any one of the n ways. Since both the operations are performed in any one of either m or n ways, why is that we get m x n ways? Here we have to understand that the first operation is performed in only one of the m ways, but with this one way we can associate n ways of doing the second operation. In other words, we have 1 x n = n ways of performing both the operations, taking into consideration not more than one way of performing the first operation. And therefore corresponding to m ways of performing the first operation we have m x n ways of performing both the operations.
Remember that this concept can be applied even if we have more than two operations. The following example should make this concept clear.
A person from his office can go to his residence via one of the 3 routes. In how many ways can that person go to his residence via one route and come to office by another route.
That person can go to his residence by one of the three routes. That is, he has 3 ways. Now he can come to office via one of the remaining two routes since he should not take the same route. That is, he can do so in two ways. Therefore, the number of ways that person can go to his residence and come back to his office by 3 x 2 = 6 ways.
Now we look at Permutations and its related concepts. Permutations are defined as each of the arrangements that can be made by taking some or all of the elements given. Here the word arrangement should be understood properly. This will be clear if we consider the given example of taking two out of four units A, B, C and D. The permutations of taking two units out of four can be done in the following ways.
AB, AC, AD, BC, BD, CD
BA, CA, DA, CB, DB, DC
Here we are looking at arranging two units in a particular order. In other words, the arrangement AB is not the same as the arrangement BA and therefore, it is necessary to list both of them. Thus AB and BA both are different arrangements of two units A and B.