What are some examples of circular permutations

Combination, variation, permutation

The following explains the difference between combination, variation and permutation. When determining the possible and favorable cases of a random experiment, you first break down the outcomes that are of interest to you into underlying elementary events and consider their arrangement. For example, if you want to know how likely it is that you will be assigned three jacks when you deal, the natural event is the receipt of a certain card.

It depends on the following points:

  • Do all elements of the sample come from the basic set?
  • Is the order or sequence of occurrence significant?
  • Have the natural events been repeated?

When playing cards, for example, it makes a difference whether you distribute all cards to the players immediately when you deal and the entire hand is in circulation at the start of the game, or whether each player receives around five cards and the remaining cards remain in the stock. Initially, the order in which the cards are dealt does not matter. However, if you wait for a particular card during the game, it is important when you receive it.

What is a permutation?

A permutation is the arrangement of n distinguishable elements in a certain order.

In the case that no repetitions occur, the number of possible permutations from n elements with n factorial is given:

   

For example, three pens (n ​​= 3) in the colors red (r), black (S) and blue (B) are randomly distributed to three people. Then there are 3! = 6 different options. As long as no pen has been distributed, there are three pens for the first person to receive. If the first pen is then taken, there are still two options for the second person. After handing out the second pen, there is only one option left for the third person:

Person 1 receivesPerson 2 receivesPerson 3 receives
R.S.B.
R.B.S.
S.R.B.
S.B.R.
B.R.S.
B.S.R.

Permutations with repetitions

In the case of permutations with repetitions, in contrast, not all elements are distinguishable. If you have n elements, of which m are identical, the number of possible different arrangements is smaller:

   

If you have two of the three pens (n ​​= 3) in black (S) and one in red (R) and you want to distribute them to three people, there are m = 2 identical objects and you only receive

   

possible different arrangements.

Person 1 receivesPerson 2 receivesPerson 3 receives
R.S.S.
S.S.R.
S.R.S.

In general, among the n objects, there are s objects, each in If repetitions occur, then the number of possible permutations is through

   

given.

What is a variation?

A variation k of n elements of the basic set is a part of the basic set, for which the order of the arrangement is also important. If all elements are distinguishable from one another, one speaks of a variation without repetition and the number of different variations of k out of n elements is:

   

From 6 different pictures ( to ), for example, you will be randomly assigned 2 pictures. With the first picture you could get each of the six pictures, with the second picture only one of the five remaining pictures. The total number of possible variations is therefore 30.
This is shown in detail in the table, the lines of which “not yet in a different order” are not relevant here.

Variations with repetitions

Do you consider against it Variations from k of n elements of the basic set with repetitions, i.e. if the elements removed during the first pass are put back again, there are now identical elements. The element removed in the first pass could ultimately also be pulled in the second pass. For each of the k withdrawals from the basic set, each of the n elements could now be selected.
Therefore the number of different variations of k is out of n elements with

   

given.

In the picture example, for example, you will receive one of the six pictures, note which one it was, return it and receive a second picture. It can then also happen that you receive the same picture twice; so there is now possible variations.

You can see this in detail in the table: Since the first picture is put back again, there are now 6 options for the second picture:

What is a combination?

A combination out of k of n elements of the basic set is finally a part of the basic set for which, in contrast to the variation, the order of the arrangement is not relevant. If all elements can be distinguished from one another, one speaks of a combination without repetition. Then the number of different combinations of k out of n elements is:

   

Above in the table of the variation without repetition, the possible arrangements of 2 out of 6 images are listed accordingly. In a third line you can also see whether this combination of images has not yet been listed in a different order. The number of "x" is therefore 15, because

   

Combination with repetitions

You consider, however Combinations with repetitions from k of n elements of the basic set, the order of the element arrangement is irrelevant, but there are identical elements. In the picture example, for example, you return the picture you received in the first round and receive a picture a second time. So theoretically every picture could be output in both rounds.

From the variations with repetitions listed above in the table, only those arrangements are relevant that have not already been observed in a different order. Furthermore, these variations are marked with an “x” in the third row. Their number is 21.
In general, the number of combinations of k results from n elements with repetitions

   

So for your example you get

   

possible arrangements.

The table finally shows you the possible numbers of permutations, variations and combinations with and without repetitions: