Fundamental principle of counting

fundamental principle of counting (PFC) is one of the number counting methods combinatorial analysis. This principle allows us to calculate the number of possible combinations with elements that can be obtained in different ways.

The PFC is a simple but very useful method, being widely used in probability problems, in determining the number of possible events.

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fundamental principle of counting

To explain more about PFC, let's use some examples.

Example 1

To go from his house to the zoo, Júlio needs to take a bus that takes him to the station and, at the station, he needs to take another bus.

Suppose there are three bus lines that take you to the station, lines A1, A2 and A3, and that there are two lines that take you from the station to the zoo, lines B1 and B2. The diagram below illustrates this situation:

In as many ways as possible Júlio can go from his house to the zoo, combining the available bus lines.

From the illustration, we can see that there are 6 possibilities in total. However, we can discover this result even without the illustration.

By PFC, we multiply the number of possible lines in the first part of the path by the number of possible lines in the second part:

From home to the station: Lines A1, A2 and A3 → 3 different ways;
From the station to the zoo: Lines B1 and B2 → 2 different ways;

$\dpi{120} \boldsymbol{3 \times 2 6}$

Example 2

In a restaurant, the customer can choose between 4 options for starters, 5 options for main course and 3 options for dessert. In how many possible ways can a customer choose a starter, main course, and dessert at this restaurant?

Prohibited: 4 options;
Main course: 5options;
Dessert: 3 options.

By the PFC, just multiply these three quantities: $\dpi{120} \boldsymbol{4 \times 5 \times 3 60}$

Therefore, there are 60 possible combinations that the customer can choose from, with a starter, a main course and a dessert in this restaurant.

Example 3

How many different words can be formed by changing the order of the letters in the word SCHOOL?

See that the letters of the word school are not repeated, they are all different. Then, in the formed words, there cannot be repeated letters either.

Considering the 6 possible positions for the letters in the word, we have:

1st position: 6 letters available;
2nd position: 5 letters available;
3rd position: 4 letters available;
4th position: 3 letters available;
5th position: 2 letters available;
6th position: 1 letter available.

By the PFC, just multiply these quantities:

$\dpi{120} \boldsymbol{6 \times 5 \times 4 \times 3 \times 2 \times 1 720}$

See how important PFC is! Without it, we would have to write down all possible words and then count them to arrive at the number 720.

Words formed from the letters of another are called anagrams.

Probability

The PFC has a lot of application in the problems of probability. The principle is used to determine the number of possible events in an experiment.

Example:

A die is thrown three times in a row and the face obtained is checked. What is the probability that there is an even face on the first toss, an odd on the second toss, and a face greater than 4 on the third toss?

Favorable cases:

1st launch: 3 possibilities (faces 2, 4 and 6);
2nd launch: 3 possibilities (faces 1, 3 and 5);
3rd launch: 2 possibilities (face 5 and 6).

By PFC, to obtain the number of favorable cases, just multiply the quantities:

$\dpi{120} \boldsymbol{3 \times 3 \times 2 18}$

Possible cases:

1st launch: 6 possibilities (faces 1, 2, 3, 4, 5 and 6);
2nd launch: 6 possibilities (faces 1, 2, 3, 4, 5 and 6);
3rd launch: 6 possibilities (faces 1, 2, 3, 4, 5 and 6).

By PFC, we can also obtain the number of possible cases:

$\dpi{120} \boldsymbol{6 \times 6\times 6 216}$

Thus, we can calculate the desired probability:

$\dpi{120} \boldsymbol{P \frac{Total \, of \, cases\, \acute{a}able}{Total \, of\, possible \ cases} \frac{18}{216} \ frac{1}{12} \approx 0.083}$

Therefore, the chance that it came up with an even face on the first toss, an odd face on the second toss and a face greater than 4 on the third toss is one in twelve, which equals approximately 0.083 or 8,3%.

Combinatorial analysis

From the PFC other techniques for counting elements are obtained: permutation, arrangement and combination.

Permutation

Allows you to calculate the number of possibilities to organize a total of n elements, changing the positions of the elements among themselves.

$\dpi{120} P_n n!$

Arrangement

It allows to calculate the number of possibilities to organize n elements in groups of size p, when the order of the elements is important within each group.

$\dpi{120} A_{n, p} \frac{n!}{(n-p)!}$

Combination

It allows to calculate the number of possibilities of organizing n elements in groups of size p, when the order of the elements no is important within each group.

$\dpi{120} C_{n, p} \frac{n!}{p!(n-p)!}$

You may also be interested:

conditional probability
Statistic
Grouping data into ranges
Dispersion measures
Mean, mode and median

Fundamental principle of counting

fundamental principle of counting

Probability

Combinatorial analysis

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