Permutation and Combination

Both the terms, i.e., Permutation and Combination, are an arrangement of a certain number of items. The main difference is that combination refers to the process in which all possibilities are tried and permutation refers to the process in which only one possibility is tried at a time. These two types of arrangements have many similarities and can be used interchangeably when working with sets containing quantities of two or fewer elements.

When both of these terms are used, it is very likely that what is being described as a combination has been created by permuting the set.

Permutation

A permutation is a type of algorithm that calculates all possible combinations in a given set of consecutive items that belong to the same group. The classic example of a permutation problem is the combination lock. This problem has two states – “unlocked” and locked.” By changing the arrangement of the locks, you can have multiple combinations in your set. Depending on how many locks are available for selection, there could be an infinite number of combinations for each state.

Example of Permutation

A permutation is a particular ordering of objects.

Example 1: The permutation (1,2,3) means that 1 precedes 2, which precedes 3. It is the same as (2,1,3) and (3,2,1). This can be extended to an ordering of objects in any number of ways. A permutation of n items has n! Possible orderings. For example, the permutation (4) has only one possible ordering: 4132.

Example 2: The permutation (1,3,7,2) implies that the one object precedes the three objects, which precedes the seven objects, which precedes the two objects. You can extend this to other orders by flipping around some of the pairs in the ordering. For example, (1,3,4) is a different permutation: 1432

Combination

A combination problem requires the input of two items. These two items must belong to the same group. For example, if you have a set of three cards and want to know which one is the fastest in a certain race, it would be impossible for you to make use of a permutation algorithm in this case because in a racing scenario, three cards never share the same group. This is due to each card representing an individual combination. 

Example of Combination

A combination is a method of getting the desired item out of a set.

Example 1: If you have ten items for sale and one person wants three items, how many ways can this be done? The answer is 5! = 6 = 2 + 3.

If we wanted all the items, how many ways can it be achieved? The answer is 2! = 4 = 2 + 1. 

Example 2: If you have ten items and you want to choose three items from them, how many ways can this be done? This requires a combination. The answer is: 10C3 = 3! = 6.

Differences between the Permutation and Combination

Following are the key differences between Permutation and Combination: 

When to use?

A permutation is used if you are given n items and you have to choose r of them out of the n.

The combination is used. If you are given k items and you want to group k things together so that all possible combinations are formed, choose one object from each group. The number of groups should be equal to the number of objects picked. In this case, it’s k = n. 

Whether they happen in groups or not groups

In permutation, if you choose an object from every group, you can get every combination. But if you choose an object from a group, then all possible combinations won’t be formed. When we pick one item randomly, we always pick one object out of the whole set, i.e., not just out of the single group that we picked it from.

Method of doing them

Permutations are done in a sequence. We always do permutations in a specific order and this means the object is fixed to a position in the line, it can’t be changed.

Combinations are done with any rotation. We can group 4 objects and then choose one of them, but we can combine them in any way we want, even if they aren’t in the order of 1,2,3 and 4.

Conclusion

Permutations can only happen in groups. Once you have done  permutation, you cannot reverse it back to its original state. In  combination, you choose one or more items (individuals) from the same group. Then, there are two things involved, choosing an element from the source set and then arranging that chosen element according to your own preference. 

Combinations are not as well known as permutations because of their simplicity and the fact that we need to calculate them using methods other than using formulas or formulae. 

But I think combinations are important and useful because they help us understand permutations better.