What is a distinguishable permutation

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Permutation with repetition

If they are indistinguishable from the elements, the result, as the following examples illustrate, are fewer than possible arrangements:

I.; Elements =.

Permutations:

Of the permutations for distinguishable elements only remain because the interchanges and for replaced by are no longer distinguishable. It applies.

II.; Elements =.

Permutations:

Of the permutations for distinguishable elements, only there remain the interchanges

for and replaced by are no longer distinguishable. It applies. We introduce the symbol for the number of permutations for the same of all elements. The two examples show that then applies

or in general

Accordingly, applies to the number of permutations with the same (or repeated) all of the elements

Our considerations can easily be extended to the case that several groups of the same elements exist. Here, too, first an example:

; Elements =.

Permutations:

The two indistinguishable 's, like the' s, result in indistinguishable permutations. Accordingly, the total number of all orders contains (i.e.) identical orders that must not be counted. We get accordingly

where now carries the double index. In general, for elements that contain r groups with indistinguishable elements in each case, the number of permutations results as

under the condition .

A chemically important question is related to the explanations here. We consider particles that can take on one of the values ​​of a certain property (e.g. absolute velocity of gas particles in the ranges or energy levels, etc.). and in the example above, 2 particles with the value or correspond. If the particles are distinguishable (i.e. numbered with the designations 1, 2, 3 and 4), then according to the above 6 arrangements arise here:

A second important case concerns the number of chemical structural isomers. As an example, consider the structural isomers of the molecule

They arise through different arrangements of the four single () and three double bonds () on the total of seven C-C bonds (with a corresponding shift of the H atoms!). In the notation, the isomer shown corresponds to. It results

Of course, this mathematical result does not mean that all isomers can actually exist.