Counting techniques: types, how to use them and examples

This statistical concept has a number of practical applications.

The world of mathematics, as fascinating as it is fascinating, is also complicated.But perhaps thanks to its complexity we can cope with day-to-day 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.

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These techniques make it possible to speed up in a very significant way to know how many different ways there are to make sequences or combinations of objects, without losing patience or sanity. Let's take a closer look at what they are and which ones are most commonly used.

Counting techniques: what are they?

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

This concept will be understood in a simpler way 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 side by side?

This problem could be solved manually, thinking of combinations such as blue, red and yellow; blue, yellow and red; red, blue and yellow, red, yellow and blue... But this may require a lot of patience and time, and for that we would make use of counting techniques, being for this case necessary a permutation.

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The five types of counting techniques

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

Actually, this type of techniques can be divided into two groups, depending on their complexity, one being made up of the multiplicative principle and the additive principle, and the other being made up of combinations and permutations.

1. Multiplicative principle

This type of counting technique, together with the additive principle, makes it easy to understand in a practical way how these mathematical methods work.

If an event, let us call it N1, can occur in several ways, and another event, N2, can occur in several other ways, then the events together can occur in N1 x N2 ways.

This principle is used when the action is sequential, i.e., it is made up of events that occur in an orderly fashion, such as building a house, choosing the dance steps in a discotheque, or the order to be followed in preparing a cake.

For example:

In a restaurant, the menu consists of a main course, a second course and dessert. Of main courses we have 4, of seconds there are 5 and of desserts there are 3.

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

Thus, 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 ways, 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 being three brands in the supermarket: A, B and C.

Chocolate A is sold in three flavors: dark, milk and white, and there is also the option without or with sugar for each of them.

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

C chocolate is sold in three flavors, dark, milk and white, with or without hazelnuts, peanuts, caramel or almonds, but all with sugar.

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

W = number of ways of selecting chocolate A.

Y = number of ways to select chocolate B.

Z = number of ways to select chocolate C.

The next step consists of 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 chocolate varieties.

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 permutation, there would be an arrangement of several elements in which their order or position is important to consider.

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

The formula that would be used would be the following: nPr = n!/(n-r)!

For example:

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

The following would be done:

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

4. Permutations with repetition

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

Taking into account that n are the available elements, some of them repeated.

All the n elements are selected.

The following formula is applied: = n!/n1!n2!...nk!

For example:

On a ship 3 red, 2 yellow and 5 green flags can be hoisted. How many different signals could be made by hoisting the 10 flags you have?

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

5. Combinations

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

The formula to apply is the following: nCr=n!/(n-r)!r!

For example:

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

En este caso, n = 10 y r = 2, así pues, aplicando la fórmula:

10C2=10!/(10-2)!2!=180 parejas distintas.

Referencias bibliográficas:

• Brualdi, R. A. (2010), Introductory Combinatorics (5th ed.), Pearson Prentice Hall.
• de 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.
• 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.