Ways of counting
Before you can calculate a probability you need to know how many cases there are, and counting them one by one is not an option when you are dealing with access codes, passwords or draws with thousands of possibilities. In 1st Bachillerato you already used these techniques intuitively; here you give them a name, a formula and a clear criterion for deciding which one to apply in each situation. It is the basis of everything that comes next in probability.
The counting principle
In a school cafeteria, the set menu allows a choice between 3 starters, 4 main courses and 2 desserts. In how many different ways can a student create a complete lunch…
In a school cafeteria, the set menu allows a choice between 3 starters, 4 main courses and 2 desserts. In how many different ways can a student create a complete lunch consisting of a starter, a main course and a dessert?
Arrangements with repetition: when repeats are allowed
A phone passcode allows digits to be repeated: after using a $7$, the $7$ is still available for the next position. In this case the counting principle simplifies because every stage has the same number of options.
To log in to a mobile app, a user must create a 4-digit security code. If the system allows numbers to be repeated and uses the digits from $0$ to $9$, how many…
To log in to a mobile app, a user must create a 4-digit security code. If the system allows numbers to be repeated and uses the digits from $0$ to $9$, how many different security codes can be generated?
To improve the security of smart lockers at a school, the system requires a $4$-digit access code. However, to avoid obvious combinations, the system imposes the…
To improve the security of smart lockers at a school, the system requires a $4$-digit access code. However, to avoid obvious combinations, the system imposes the following conditions: the first digit must be odd, the last digit must be even (considering $0$ as even), and no digit may be repeated in the code. How many different codes can be generated following these rules?
Arrangements without repetition
If instead of a code it is a podium (1st, 2nd, 3rd) with different competitors, each position already filled is no longer available for the following ones: the options keep getting smaller.
A group of 10 friends decides to organise a video game tournament. Prizes will be awarded to the first, second and third place finishers. Given that there can be no…
A group of 10 friends decides to organise a video game tournament. Prizes will be awarded to the first, second and third place finishers. Given that there can be no ties, in how many different ways can these three trophies be awarded?
A streetwear brand wants to launch a limited edition of hoodies. They have $3$ identical red hoodies, $2$ identical blue hoodies and $4$ identical black hoodies. For a…
A streetwear brand wants to launch a limited edition of hoodies. They have $3$ identical red hoodies, $2$ identical blue hoodies and $4$ identical black hoodies. For a photoshoot, they must place all these hoodies in a row on a counter. In how many different ways can the hoodies be arranged based solely on their colour?
Permutations: ordering all the elements
If instead of choosing $k$ out of $n$ elements you order all of them, you are dealing with a particular case of arrangement without repetition with $k=n$.
A bookshelf has space to fit exactly 6 textbooks from different Year 2 Bachillerato subjects. In how many different ways can these 6 books be arranged on the shelf if…
A bookshelf has space to fit exactly 6 textbooks from different Year 2 Bachillerato subjects. In how many different ways can these 6 books be arranged on the shelf if they must all be aligned next to each other?
In a regional E-sports competition (competitive video games), $12$ teams are participating. The organisers must award a gold, silver and bronze medal to the top three…
In a regional E-sports competition (competitive video games), $12$ teams are participating. The organisers must award a gold, silver and bronze medal to the top three teams.
- a) In how many different ways can the competition podium be formed?
- b) If we know that the local team "Los Gamers" always finishes in the top three, in how many ways can the podium be completed now?
Permutations with repeated elements
What happens if some of the elements you are ordering are indistinguishable from each other, such as two identical letters in a word? Some of the $n!$ orderings turn out to be identical, so you have to divide so as not to count them more than once.
A group of $15$ students in $2º$ Baccalaureate is organising the graduation party. They must choose a committee formed of $4$ people to be in charge of the music and…
A group of $15$ students in $2º$ Baccalaureate is organising the graduation party. They must choose a committee formed of $4$ people to be in charge of the music and food.
- a) How many different committees can be formed?
- b) If there are $8$ girls and $7$ boys in the group, how many of those committees would be formed of $2$ girls and $2$ boys?
An architecture firm is designing a housing estate with 7 identical plots in a row. They must plant trees of three different species: oak, pine and holm oak. For…
An architecture firm is designing a housing estate with 7 identical plots in a row. They must plant trees of three different species: oak, pine and holm oak. For aesthetic and biological reasons, the following requirements must be met:
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Only one tree is planted in each plot.
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There must be at least one tree of each species.
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No two pine trees can be in adjacent plots.
Calculate the total number of different ways in which the trees can be distributed among the 7 plots.