Counting techniques
Laplace's rule, $P(A) = \frac{\text{favorables}}{\text{posibles}}$, is easy to state but sometimes very hard to apply: how many different menus are there? In how many ways can three prizes be shared among eight players? Counting one by one becomes impossible as soon as the numbers grow. This section gives you the tools to count without listing: the multiplication principle and the two most common types of counting, depending on whether or not order matters.
Tree diagrams and the multiplication principle
A streaming* platform lets you customize your user profile by choosing an avatar from 4 different animals (lion, panda, cat and wolf) and a background color from 3…
A streaming platform lets you customize your user profile by choosing an avatar from 4 different animals (lion, panda, cat and wolf) and a background color from 3 options (blue, green and orange).
- a) Describe the sample space $E$ of the possible profile combinations.
- b) How many elements does the sample space have? Use the multiplication principle to justify it.
In your school cafeteria menu you can choose a starter between salad, pasta or rice; a main course between chicken or fish; and a dessert between fruit or yoghurt. a)…
In your school cafeteria menu you can choose a starter between salad, pasta or rice; a main course between chicken or fish; and a dessert between fruit or yoghurt.
- a) Represent all possible complete menu options using a tree diagram.
- b) If you choose a menu at random, what is the probability that it includes "pasta" as a starter?
Permutations of selections: choosing and ordering
When several elements are chosen and ordered without repetition (a podium, a set of different posts), the count is called a permutation of a selection (known in Spanish as a variación). When repetition is allowed (a code, a password), it is a permutation with repetition.
A combination padlock for a locker has a 3-digit code. Each digit can be a number from $1$ to $5$. a) How many different combinations can be created if numbers can be…
A combination padlock for a locker has a 3-digit code. Each digit can be a number from $1$ to $5$.
- a) How many different combinations can be created if numbers can be repeated?
- b) How many combinations are there if the three numbers must be distinct?
In a local chess tournament 8 players participate. Three different trophies will be awarded for first, second and third place. a) In how many different ways can the…
In a local chess tournament 8 players participate. Three different trophies will be awarded for first, second and third place.
- a) In how many different ways can the three prizes be awarded?
- b) If last year's champion participates again and we are certain he will finish in the top three, how many podium finishes are possible now?
Combinations: choosing without ordering
When several elements are chosen and the order does not matter (a committee, a selection of songs, a hand of cards), the count is called a combination and is calculated with the binomial coefficient $\binom{n}{k}$, read "$n$ choose $k$".
A band has prepared 10 songs for their new album, but the record label tells them they can only include 4 of them. a) How many different selections of 4 songs can they…
A band has prepared 10 songs for their new album, but the record label tells them they can only include 4 of them.
- a) How many different selections of 4 songs can they make if the order does not matter in which they are chosen?
- b) What are these types of groupings called in combinatorics? Justify your answer.
A streaming platform uses a security code for parental control. The code consists of 2 letters (from a 26-letter alphabet, without Ñ) followed by 3 digits (from 0 to 9).…
A streaming platform uses a security code for parental control. The code consists of 2 letters (from a 26-letter alphabet, without Ñ) followed by 3 digits (from 0 to 9).
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a) How many different codes can be created if letters and numbers can be repeated?
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b) If a user forgets their code but remembers that it begins with the letter "A" and ends with the number "7", how many possible combinations would they have to try?
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c) If the code were generated at random without allowing any repeated characters (letters or numbers), what would be the probability of the code being "BX123"?
Draws with and without replacement
A survey was conducted among 200 young people about their reading and television viewing habits. The data are shown in the following contingency table: | | Reads…
A survey was conducted among 200 young people about their reading and television viewing habits. The data are shown in the following contingency table:
| Reads regularly | Does not read regularly | Total | |
|---|---|---|---|
| Watches TV > 2 h/day | 35 | 85 | 120 |
| Watches TV < 2 h/day | 55 | 25 | 80 |
| Total | 90 | 110 | 200 |
If a young person is chosen at random from this survey, calculate:
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a) The probability that they read regularly and watch television less than 2 hours a day.
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b) Given that the young person reads regularly, what is the probability that they watch television more than 2 hours a day?
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c) Are the events "Reads regularly" and "Watches television more than 2 hours a day" independent? Justify your answer mathematically.
A logistics company organizes its routes using a barcode of $10$ vertical stripes. Each stripe can be black, grey or white. For scanner readability reasons, codes with…
A logistics company organizes its routes using a barcode of $10$ vertical stripes. Each stripe can be black, grey or white. For scanner readability reasons, codes with three consecutive stripes of the same colour are not allowed.
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How many different codes of $10$ stripes can be formed in total (without restrictions)?
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If a code is chosen at random from all possible codes (without restrictions), what is the probability that the stripes in the odd positions are all black and those in the even positions are all white or grey?