Permutations vs Combinations
Permutation and Combination are two closely related concepts. Though they appear to be out from similar origin they have their own significance. In general both the disciplines are related to ‘Arrangements of objects’. However slight difference makes each constraint applicable in different situations.
Just from the word ‘Combination’ you get an idea of what it is about ‘Combining Things’ or to be specific: ‘Selecting several objects out of a large group’. At this particular point of situation finding the Combinations does not focus on ‘Patterns’ or ‘Orders’. This can be clearly explained in this following example.
In a tournament, no matter how two teams are listed unless they clash between them in an encounter. It doesn’t make any difference, if team ‘X’ plays with team ‘Y’ or team ‘Y’ plays with team ‘X’. Both are similar and what matters is both get the chance to play against each of the other regardless of the order. Thus a good example to explain the combination is making a team of ‘k’ number of players out of ‘n’ number of available players.
nk (or n_k ) = n!/k!(n-k)! is the equation used to compute values for a common ‘Combination’ based problem.
On the other hand ‘Permutation’ is all about standing tall on ‘Order’. In other words the arrangement or pattern matters in permutation. Therefore one can simply say that permutation comes when ‘Sequence’ matters. That also indicates when compared to the ‘Combination’, ‘Permutation’ has higher numerical value as it entertains the sequence. A very simple example that can be used to clearly bring the picture of ‘Permutation’ is forming a 4 digit number using the digits 1,2,3,4.
A group of 5 students are getting ready to take a photo for their annual gathering. They sit in ascending order (1, 2, 3, 4, and 5) and for another photo, the last two inter-change their seats mutually. Since the order is now (1, 2, 3, 5 and 4) which is entirely different from the aforementioned order.
nk (or n^k) = n!/(n-k)! is the equation applied to calculate ‘Permutation’ oriented questions.
It is important to understand the difference between permutation and combination to easily identify the right parameter that has to be used in different situations and to solve the given problem. In common, ‘Permutation’ results higher in value as we can see,
n^k = k! (n_k) is the relativity between them. In norm, questions carry more ‘Combination’ problems since they are unique in nature.