Counting techniques: types, how to use them and examples

The world of mathematics, just as fascinating is also complicated, but perhaps thanks to its complexity we can cope with the day-to-day more effectively and efficiently.

Counting techniques are mathematical methods that allow us to know how many different combinations or options there are 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 knowing 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 the most 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 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 easily 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 arranged side by side?

This problem could be solved by doing it 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 use counting techniques, for this case a permutation is 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 with its own peculiarities 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, 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 way, such as the construction of a house, choosing the dance steps in a disco or the order that will be followed to prepare a pie.

For example:

In a restaurant, the menu consists of a main course, a second and dessert. For main dishes we have 4, for seconds there are 5 and for 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.

This means that if the first activity can occur in M ​​ways, the second in N and the third 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, in addition to having the option without or with sugar for each of them.

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

Chocolate C is sold in three flavors, black, milk and white, with the option of having 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 bought?

W = number of ways to select the chocolate A.

Y = number of ways to select the chocolate B.

Z = number of ways to select the chocolate C.

The next step is 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 It has a series of steps to be carried out, 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 the 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.

On the other hand, in a permutation, there would be an arrangement of several elements in which it is important to take into account their order or position.

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

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 that can only fit five, how many ways can they sit?

The following would be done:

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

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:

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

All items n are selected.

The following formula applies: = n! / N1! N2... nk!

For example:

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

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

5. Combinations

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

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

For example:

A group of 10 people want to clean up the neighborhood and are preparing 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. TO. (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.
• 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.