# Counting techniques: types, how to use them, and examples

The world of mathematics, fascinating as it is, is also complicatedBut maybe thanks to its complexity, we can cope with the day to day more effectively and efficiently.

Counting techniques are mathematical methods that let you know how many different combinations or options you have of items in the same group of objects.

These techniques make it very easy to know the number of different ways to create sequences or combinations of objects, without losing patience or common sense. Let’s take a closer look at what they are and which are used the most.

## Counting techniques: what are they?

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

This concept is more easily understood 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 sorted 1:00 am on the other 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 it can take a lot of patience and time, and that’s why we would use counting techniques, being for this case a permutation is necessary.

## The five types of counting techniques

The main counting techniques are the five, Although these are not the only ones, each with its own peculiarities and used as needed to find out how many combinations of object sets are possible.

Indeed, these types of techniques can be divided into two groups, depending on their complexity, one consisting of the multiplier principle and the additive principle, and the other consisting of combinations and permutations.

### 1. Multiplicative principle

This type of counting technique, associated with the additive principle, makes it easy and practical to understand the operation of these mathematical methods.

If an event, say-N1, can occur in more than one way and another event, N2, can occur in as many, then the events together can occur in the forms N1 x N2.

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 the construction of a house, the choice of dance steps in a box night or the order in which will continue to bake a cake.

For example:

In a restaurant, the menu consists of a main course, a main course and a dessert. We have 4 main courses, 5 main courses and 3 main courses.

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

The combinations offered by this menu would therefore be 4 x 5 x 3 = 60

### 2. Additive principles

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

This means that if the first activity can occur in the M forms, the second of 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: dark, with milk and white, in addition to having the option without or with sugar for each of them.

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

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

On this basis, the question to be answered is: how many different varieties of chocolate can you buy?

W = number of ways to select chocolate A.

I = number of ways to select chocolate B.

Z = number of ways to select chocolate C.

The next step is a simple multiplication.

W = 3 x 2 = 6.

I = 3 x 2 x 2 = 12.

Z = 3 x 5 = 15.

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

To know whether the multiplicative principle or the additive should be used, the main clue is whether the activity in question has a series of steps to be carried out, as was the case with the menu, or whether there is several options, as is the case with Chocolat.

### 3. Permutations

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

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

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

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

The formula that would be used would be: NPR = n! / (No)!

For example:

There is a group of 10 people and there is a seat that only 5 people can fit in, 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 of occupying the bench.

### 4. Permutations with repetition

When you want to know the number of permutations in a set of objects, some of which are equal, do the following:

Since n are the available items, some of them are repeated.

All n elements are selected.

The formula applies: = n! / N1! N 2! … nk!

For example:

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

ten! / 3! 2! 5! = 2520 different flag combinations.

### 5. Combinations

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

The formula to apply is: NCR = n! / (No)! R!

For example:

A group of 10 people wants 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, so by applying the formula:

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

#### Bibliographical references:

• Brualdi, RA (2010), Introductory Combinatorics (5th ed.), Pearson Prentice Hall.
• de Finetti, B. (1970). “Logical foundations and the measure of subjective probability”. Acta Psychologica.
• Hogg, RV; Craig, Allen; McKean, Joseph W. (2004). Introduction to mathematical statistics (6th ed.). Upper Saddle River: Pearson.
• Mazur, DR (2010), Combinatorics: A Guided Tour, Mathematical Association of America,
• Ryser, HJ (1963), Combinatorial Mathematics, The Mathematical Monographs of Carus 14, Mathematical Association of America.