With this in mind, we define permutation in the following way: Take a look at the figure below to see the process of arranging only two out of four boxes from the set above, and see how the amount of different possible ways for this changes.įigure 3: Four boxes, two to be arranged side by side, and the result of all the possible arrangements This is exactly what permutations are all about! To find all the possible ways in which a group of items from a set (could be all of the items from the set, or could be just some of them) can be arranged side by side. Notice the process is EXACTLY the same as before, but with less possible positions! Therefore, we multiply the possible ways for each position and obtain 4 × 3 = 12, for a total of 12 possible different arrangements of two boxes out of a set of four. This time, you have four boxes to be arranged in two positions, therefore, you know that in the first position you have 4 possible boxes to be put in there while in the second position, you are left with three possible boxes to arrange since you have already used one.

So we have imposed a condition: only two out of the four boxes will be arranged, but notice that we didn't say WHICH boxes! Therefore, all of the four boxes are still up for the arrangement, is just that you can only use two at a time. Now let us keep those same 4 boxes, but we will order ONLY TWO of them and see how many different ways we can arrange them. This is a type of permutation calculation because each arrangement is defined by the order in which the boxes have been ordered this is also the simplest type of permutation calculation, since you have four items, all of them to be ordered in different ways (the number of items, is equal to the number of possible positions). When multiplying the number of possible boxes for each of the positions, you will find all the ways in which the arrangements of the boxes can be done, which is equal to 4! = 4 × 3 × 2 × 1 = 24 (24 different ways). Pick one once more, move to the next position where you will be left with two possible boxes, pick one and then for the last position you only have one possible box left.

$permutations and combinations examples$

We saw this process before, we know that in the first position you have four possible boxes to be used, therefore you pick one and move to the next position, where you will have only three possible boxes to choose (having used one in the first position). How many possible different arrangements are there for these four boxes, when ALL of them will be arranged side by side in different ways? Well, the answer is very simple, is just 4!įigure 2: Four boxes to be arranged side by side, and the result of all the possible arrangements Think on the next example (very similar to what we have seen so far): we have four boxes (A, B, C and D) and we will arrange them side by side.įigure 1: Four boxes to be arranged side by side You may be wondering then, how is this different from the method to compute all the possible arrangements of a list of items side by side that we have seen before? Well, in simple words, when all of the objects from the set will be arranged side by side, permutations produce the total quantity exactly in the same way as we have calculated it before, the difference comes when you have a list of items and you will not be arranging them all. A permutation refers to this order and all of the possible arrangements of the items according to specific conditions, which limit the amount of items from the set that will be used in the arrangements. Let us explain, we have seen example problems where we had to find out all of the different possible arrangements of items from a set that are ordered side by side. If you think about it, it sounds exactly the same as what we have seen before in our lesson about factorials, because it actually is in specific cases!

The idea of permutation is rooted in the process of arranging and finding how many total possible arrangements exist for a group of items in a set. Without further ado, let us start by introducing permutations and combinations separately. Therefore, we have decided to dedicate this lesson to introduce both concepts together, the permutation and the combination, and later, we will have a lesson dedicated to each one and a variety of example problems to practice them further. After our lesson on factorials we are ready to learn about the most important part of combinatorics: permutations and combinations in reality, what we learned about the factorial notation in our past lesson (please make sure you read that lesson first!) provides the basis for these two new concepts, which are deeply intertwined with each other.

Permutations and combinations examples