A permutation arranges all the elements of a set. But many situations are different: choosing and arranging only some of them. In how many ways can the gold, silver and bronze medals be awarded among 8 athletes? You are not arranging all 8, just choosing 3 and placing them in 3 positions. That is a variation.
You reason in the same way as with permutations, but you stop at the $k$-th factor instead of going all the way down to 1: for the first position there are $n$ options, for the second $n-1$, and so on until position $k$, where there are $n-k+1$ options left.
In an art gallery, $2$ paintings selected from a private collection of $7$ works are to be displayed. One will be placed on the left wall of the room and the other on…
In an art gallery, $2$ paintings selected from a private collection of $7$ works are to be displayed. One will be placed on the left wall of the room and the other on the right wall. In how many different ways can this exhibition be arranged?
A book club has $9$ members. They need to choose a committee consisting of a chairperson, a vice-chairperson, and a secretary. As the positions are distinct and each…
A book club has $9$ members. They need to choose a committee consisting of a chairperson, a vice-chairperson, and a secretary. As the positions are distinct and each person can only hold one, in how many ways can the committee be formed?
What if the same element can appear more than once in the sequence? When you roll a die 4 times in a row, the result of each roll does not depend on the previous ones: a "6" can come up all 4 times. You no longer remove options at each stage.
$$VR_{n,k} = n^k$$
Each of the $k$ positions always has the same $n$ options available, because nothing is "used up": it is the multiplication principle with the same factor repeated $k$ times.
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We roll a traditional $6$-sided cubic die $4$ consecutive times and record the result of each roll in order. How many different outcomes can we obtain in total?
We roll a traditional $6$-sided cubic die $4$ consecutive times and record the result of each roll in order. How many different outcomes can we obtain in total?
We want to paint $3$ postboxes lined up in a street. We have $4$ different colours available (red, green, blue, and yellow). If we can repeat colours on different…
We want to paint $3$ postboxes lined up in a street. We have $4$ different colours available (red, green, blue, and yellow). If we can repeat colours on different postboxes, in how many different ways can we paint them?
A country's number plate system consists of a block of $4$ digits (from $0$ to $9$) followed by a block of $2$ letters chosen from an alphabet of $20$ consonants. In…
A country's number plate system consists of a block of $4$ digits (from $0$ to $9$) followed by a block of $2$ letters chosen from an alphabet of $20$ consonants. In both blocks, repetition of characters is permitted. How many different number plates can be issued in total with this format?
To access a secure database, a $5$-character password is required. For security reasons, the system requires the first two characters to be vowels (repetition is…
To access a secure database, a $5$-character password is required. For security reasons, the system requires the first two characters to be vowels (repetition is allowed) and the last three characters to be numbers from $0$ to $9$ (with none repeated). How many different passwords satisfy this format?
In inverse problems, you know the result of $V_{n,k}$ or $VR_{n,k}$ and have to find $n$. It is usually enough to expand the formula and solve the resulting equation, trying small values if necessary.
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In a chess tournament, a double round-robin is played (each player faces all the others twice, alternating the colour of the pieces). If a total of $90$ games were…
In a chess tournament, a double round-robin is played (each player faces all the others twice, alternating the colour of the pieces). If a total of $90$ games were played, set up the corresponding equation using the formula for variations (arrangements) and find how many players participated in the tournament.
Given that the number of variations with repetition of $x$ elements taken $2$ by $2$ exceeds by $12$ the number of variations without repetition of the same $x$ elements…
Given that the number of variations with repetition of $x$ elements taken $2$ by $2$ exceeds by $12$ the number of variations without repetition of the same $x$ elements taken $2$ by $2$, set up the corresponding mathematical equation and find the value of $x$.