Combinatorics
In the previous section you counted cases "by hand", reasoning each time about whether order mattered or elements were repeated. Combinatorics gives each of those situations a name and a fixed formula, so you don't have to reinvent the reasoning every time: you only need to identify what type of count it is.
The multiplication principle and arrangements with repetition
A restaurant offers $4$ starters, $5$ main courses and $3$ desserts on its set menu. Calculate how many different complete meals (consisting of a starter, a main course…
A restaurant offers $4$ starters, $5$ main courses and $3$ desserts on its set menu. Calculate how many different complete meals (consisting of a starter, a main course and a dessert) a customer can order.
A digital bicycle lock is opened using a $4$-digit code (from $0$ to $9$). Determine how many different codes can be set if repeating digits is allowed.
A digital bicycle lock is opened using a $4$-digit code (from $0$ to $9$). Determine how many different codes can be set if repeating digits is allowed.
Arrangements without repetition: choosing and ordering
A robotics club has $12$ members. Determine how many different ways a committee consisting of a president, a vice president and a secretary can be chosen, given that no…
A robotics club has $12$ members. Determine how many different ways a committee consisting of a president, a vice president and a secretary can be chosen, given that no person can hold more than one position.
A security system requires a password consisting of $2$ distinct vowels followed by $3$ distinct digits (from $0$ to $9$). Calculate how many different passwords can be…
A security system requires a password consisting of $2$ distinct vowels followed by $3$ distinct digits (from $0$ to $9$). Calculate how many different passwords can be configured under these conditions.
Permutations: ordering all the elements
Calculate how many different ways $6$ different textbooks can be arranged in a line on a shelf.
Calculate how many different ways $6$ different textbooks can be arranged in a line on a shelf.
Calculate the number of different anagrams, whether meaningful or not, that can be formed using all the letters of the word $INGENIERIA$.
Calculate the number of different anagrams, whether meaningful or not, that can be formed using all the letters of the word $INGENIERIA$.
Combinations: choosing without ordering
You already know the binomial coefficient $\binom{n}{k}$ from the binomial theorem: in combinatorics it is called a combination and it counts the ways of choosing $k$ elements from $n$ when order does not matter.
A technology teacher needs to select $4$ students from a class of $20$ to attend a science fair. Calculate how many different groups of representatives can be formed.
A technology teacher needs to select $4$ students from a class of $20$ to attend a science fair. Calculate how many different groups of representatives can be formed.
$8$ points are marked on a plane, no three of which are collinear. Determine how many different straight lines can be drawn by connecting pairs of these points.
$8$ points are marked on a plane, no three of which are collinear. Determine how many different straight lines can be drawn by connecting pairs of these points.
From a Spanish deck of $40$ cards, $3$ cards are drawn at random without replacement. Calculate the probability that all three cards are Kings.
From a Spanish deck of $40$ cards, $3$ cards are drawn at random without replacement. Calculate the probability that all three cards are Kings.
Combinatorics with several groups and restrictions
In real problems it is useful to combine techniques: choosing $k$ from one group and $j$ from another (multiplying combinations), requiring some elements to stay together, or distributing elements among several destinations.
A biomedical research team must consist of $5$ people. The available candidates are $6$ biologists and $4$ chemists. Determine how many different teams can be formed if…
A biomedical research team must consist of $5$ people. The available candidates are $6$ biologists and $4$ chemists. Determine how many different teams can be formed if the team must contain exactly $3$ biologists and $2$ chemists.
Seven friends go to the cinema and sit in a row of $7$ seats. Two of them, Ana and Bruno, must sit next to each other. Calculate how many different ways they can occupy…
Seven friends go to the cinema and sit in a row of $7$ seats. Two of them, Ana and Bruno, must sit next to each other. Calculate how many different ways they can occupy the seats.