The world of mathematics, as well as being fascinating, is also complicated , but perhaps thanks to its complexity we can cope with everyday life more effectively and efficiently.

Counting techniques are mathematical methods that allow us to know how many different combinations or options we have of the elements within the same group of objects.

These techniques make it possible to speed up in a very significant way the knowledge of how many different ways there are to make sequences or combinations of objects, without losing patience or sanity. Let’s see more in depth what they are and which are the most used.

Counting techniques: what are they?

Counting techniques are mathematical strategies used in probability and statistics that allow to determine the total number of results that can exist from making combinations within a set or sets of objects. These types of techniques are used when it is practically impossible or too heavy to manually make combinations of different elements and to know how many of them are possible.

This concept will be understood more simply through an example . If you have four chairs, one yellow, one red, one blue and one green, how many combinations of three of them can be made in order next to each other?

This problem could be solved by doing it manually, thinking of combinations like blue, red and yellow; blue, yellow and red; red, blue and yellow, red, yellow and blue… But this can require a lot of patience and time, and for that we would make use of the counting techniques, being for this case a permutation necessary.

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The five types of counting techniques

The main counting techniques are the following five , although not the only ones, each one with its own particularities and used according to the requirements to know how many combinations of sets of objects are possible.

Actually, this kind of techniques can be divided into two groups, according to their complexity, being one conformed by the multiplicative principle and the additive principle, and the other, being conformed by the combinations and permutations.

1. Multiplicative principle

This type of counting technique, together with the additive principle, allows an easy and practical understanding of how these mathematical methods work.

If one event, let’s call it N1, can occur in several ways, and another event, N2, can occur in as many ways, then the events together can occur in N1 x N2 ways.

This principle is used when the action is sequential, that is, it is made up of events that occur in an orderly fashion, such as building a house, choosing dance steps in a nightclub or the order in which a cake is prepared.

For example:

In a restaurant, the menu consists of a main course, a second course and dessert. There are 4 main courses, 5 main courses and 3 desserts.

So, N1 = 4; N2 = 5 and N3 = 3.

So, the combinations offered by this menu would be 4 x 5 x 3 = 60

2. Additive principle

In this case, instead of multiplying the alternatives for each event, what happens is that the various ways in which they can occur are added together.

This means that if the first activity can occur in M forms, the second in N and the third in L, then, according to this principle, it would be M + N + L.

For example:

We want to buy chocolate, there are three brands in the supermarket: A, B and C.

Chocolate A is sold in three flavors: black, milk and white, and there is a choice of sugar or no sugar for each.

The B chocolate is sold in three flavors, black, with milk or white, with the option of having or not hazelnuts and with or without sugar.

The C chocolate is sold in three flavors, black, with milk and white, with the option of having or not hazelnuts, peanuts, caramel or almonds, but all with sugar.

Based on this, the question is: how many different varieties of chocolate can be bought?

W = number of ways to select the chocolate A.

Y = number of ways to select the B chocolate.

Z = number of ways to select the chocolate C.

The next step is a simple multiplication.

W = 3 x 2 = 6.

Y = 3 x 2 x 2 = 12.

Z = 3 x 5 = 15.

W + Y + Z = 6 + 12 + 15 = 33 different varieties of chocolate

To know whether to use the multiplicative or the additive principle, the main clue is whether the activity in question has a series of steps to be performed, as was the case with the menu, or there are several options, as is the case with chocolate.

3. Permutations

Before understanding how to do permutations, it is important to understand the difference between a combination and a permutation.

A combination is an arrangement of elements whose order is not important or does not change the final result.

In contrast, in a swap, there would be an arrangement of several elements in which it is important to take into account their order or position.

In permutations, there is n number of different elements and a number of them is selected, which would be r.

The formula to be used would be: nPr = n!/(n-r)!

For example:

There is a group of 10 people and there is a seat that can only fit five, how many ways can they sit?

Here’s what you would do:

10P5=10!/(10-5)!=10 x 9 x 8 x 7 x 6 = 30,240 different ways to occupy the bench.

4. Permutations with repetition

When you want to know the number of permutations in a set of objects, some of which are the same, you proceed as follows

Bearing in mind that n are the elements available, some of them repeated.

All elements n. are selected

The following formula applies: = n!/n1!n2!…nk!

For example:

On a boat you can raise 3 red flags, 2 yellow flags and 5 green flags. How many different signals could you make by raising the 10 flags you have?

10!/3!2!5! = 2,520 different flag combinations

5. Combinations

In combinations, unlike permutations, the order of the elements is not important.

The formula to be applied is: nCr=n!/(n-r)!r!

For example:

A group of 10 people want to clean up the neighborhood and prepare to form groups of 2 members each. How many groups are possible?

In this case, n = 10 and r = 2, thus applying the formula:

10C2=10!/(10-2)!2!=180 different pairs.

Bibliographic references:

  • Brualdi, R. A. (2010), Introductory Combinatorics (5th ed.), Pearson Prentice Hall.
  • by Finetti, B. (1970). “Logical foundations and measurement of subjective probability”. Acta Psychologica.
  • Hogg, R. V.; Craig, Allen; McKean, Joseph W. (2004). Introduction to Mathematical Statistics (6th ed.). Upper Saddle River: Pearson.

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  • Mazur, D. R. (2010), Combinatorics: A Guided Tour, Mathematical Association of America,
  • Ryser, H. J. (1963), Combinatorial Mathematics, The Carus Mathematical Monographs 14, Mathematical Association of America.