Hello again friends of steemit
For this post I continue with the explanation of the theory of combinations previously discussed from variations, for those interested in the subject I invite you to see combinations-permutation. Recall that permutations are variations of m elements taken m to m, to form groups according to the present conditions:
And for such processes we conclude to write the permutations in their mathematical expression in the following way:
Now to define combinations groups are still forming, but under other conditions described as follows:
Given m elements, we will call domain combinations n with m
n the different groups that can be formed taking n of the m given elements, so that any two of these groups are considered different when they differ in at least one of their elements.
As you see differences in the variations in this process, it does not matter the order in which the elements are arranged. For example abcd and cadb are different variations but the same combination; only a set that contains some other element, for example, abde, is considered as a new combination.
To better understand how combinations are formed, let's take, for example and again the four letters: a, b, c, d, and write their combinations of first order, second, third and fourth. Then it turns out:
For the first order are:
For the second order, the following letters are added successively to those of the first order:
The third order combinations are obtained by adding to each of the second order the letters that follow it in succession, one by one: it looks like this:
It can be noted that the combinations ad, bd and cd, are not followed by any letter in succession, therefore there are no third-order combinations for these. The fourth order are obtained by adding the letters that follow in succession, one by one, so that we have:
It is clear that the combinations _ abd_, acd and bcd do not follow any letter in succession and therefore are discarded, then with these same observations you can make examples similar, then in general, we can say that:
The number of combinations of order n that can be formed with m elements is designated by
In effect we consider the set formed by the elements a, b, c and d. If we form all the binary variations with these four elements, and we order them by verticals, in such a way that permutations of the same elements appear in each vertical, we have:
The number of these variations is
from where:
In general, each combination of order n gives rise to
Proceeding analogously with similar examples, the following can generally be established:
or replacing
that is, the number of combinations of order n of m elements is equal to the quotient of dividing the number of variations
For example:
In order to give another expression to the combinations and which is well known, multiply the two terms of the fraction by the number:
so, we have:
But if we look at the numerator of this new fraction it is equal to
Formula, which by accepting that
For example, the above expression is useful for solving problems of the type:
Analysis: By geometry it is known that two points determine straight and only one; the order of the points does not matter, because they always determine the same line. The problem is then combinations of the 7 points taken 2 to 2. So, we have:
21 lines can be formed by joining 7 points of a plane.
Analysis: If they were never in a straight line, the number of lines that can be drawn would be:
That is,
as you can see 61 lines can be formed.
This post is part of an analysis of the combinations of objects, elements or anything that can be expressed through numbers, and explained here as detailed as possible. Thank you for taking the time to see this little content and I hope it is useful or of interest to you.
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