All about combinations and permutations

The permutations and combinations are all about arranging the things in possible groupings. The elements are organized into subsets, the only difference between these two is the fact that permutation involves ordered arrangement of lists. On the other hand, combinations are used for sorting groups (where order isn’t necessary). 

We will try to explain the difference using an example:

Assume an ordered arrangement of three brothers based on decreasing order of age; Ali, Faisal and Salman. Now, in this way it depicts the permutation (because of the age factor). 

In case of combination, it doesn’t matter which comes first. It could be Faisal, Salman and Ali or we can write it as Salman, Ali and Faisal. In fact there are 6 possible combinations.

According to the above statement; a lock code that is commonly referred to as combination lock, isn’t really a combination, actually it is a permutation. Because, it involves a specific pattern or arrangement of numbers to open the lock.

Now, we will discuss these two in detail, and also devise ways to calculate these factors.

Permutations:

Let’s begin our journey with permutations, or all possible ways of doing something, keeping in mind the categorization or sequence of listing items.

For instance, we have to give 3 medals among a group of 10 students. The important thing is to keep track of the sequence. The sequence is A to J, how many ways to award 1st, 2^nd and 3^rd price among these ten participants?

To solve this, we will first break it down into three categories:

1.Gold ——– (A,B,C,D,E,F,G,H,I,J) assume that A got the first prize

2.Silver  ——- (B,C,D,E,F,G,H,I,J) assume , B won the 2^nd prize, 8 are left

3. Bronze —— (C, D, E, F, G, H, I, J) last prize won by C.

We selected certain contestants to win, but the particulars aren’t important: the main point is we had 10 sets at the start, then 9 and in the end we were left with 7. Now, if we look at the total options, 10*9*8 = 720.

We started with 10 and then eliminated the top three prize holders one by one, until we ran out of awards. In this way we get rid of dealing with too many possibilities at a time.

Let’s look at the factorial:

                            10! = 10*9*8*7*6*5*4*3*2*1

Unluckily, it would result in too many options. We only need 10*9*8, Here, permutation is helpful. Note that we want to get rid of 7 factorials.

If we try this:  10! / (10 – 3)! (Because we eliminated 3 contestants by awarding medals).  

                      10! / 7! =   10*9*8*7*6*5*4*3*2*1 / 7*6*5*4*3*2*1

                                   =    10*9*8

This gives us the formula for permutation:

                      P (n, k) = n! / (n-k)!

Here, you have n that represents the number of items, and k denotes the number of ways to arrange those items. An online permutation calculator may be used to save a considerable amount of time in case you do not want to calculate permutations manually.

Combinations:

As we discussed earlier, combinations are easy to work with, you can arrange the items in groupings in any possible pattern.

The formula is expressed as:

If we apply this to the previous example, this formula will assist us in finding all the possibilities to select k people from n, and then divide by the k!

This gives us;

Now, from the previous example, we have to award three prizes, 3! = 6 variations for each choice we select. To find combinations, divide all the permutations by these variations. We had 720 permutations, divide it by 6, 720/6 = 120 options.We hope this article will help you in understanding the concepts and calculations of these combinations, which are useful in our daily life. If you are interested to find easy methods to compute the above said problem, try an online combination calculator.