Permutations
In how many ways can 7 runners reach the finish line, with no ties? This is a question about ordering: placing all the elements of a set one after another. This is called a permutation, and it is the simplest (and most widely used) case in combinatorics.
Permutations of n distinct elements
For the first place there are $n$ options; once that is chosen, $n-1$ remain for the second; $n-2$ for the third; and so on until only $1$ option is left for the last place.
$$P_n = n \cdot (n-1) \cdot (n-2) \cdots 2 \cdot 1 = n!$$A group of $6$ friends goes to the theatre and buys tickets for consecutive seats in the same row. In how many different ways can they sit to enjoy the play?
A group of $6$ friends goes to the theatre and buys tickets for consecutive seats in the same row. In how many different ways can they sit to enjoy the play?
During a school athletics competition, $7$ runners reach the finish line with no ties. How many different finish orders can there be at the end of the race?
During a school athletics competition, $7$ runners reach the finish line with no ties. How many different finish orders can there be at the end of the race?
Permutations with position restrictions
Often part of the arrangement is fixed in advance: two people who must sit together, an element that must go first or last... In such cases you do not apply $n!$ directly: you must first reason out the restriction and then arrange what is left free.
Five friends go to the cinema and sit in adjacent seats. Two of them, Amina and Unai, want to sit together to share popcorn. In how many ways can the five friends be…
Five friends go to the cinema and sit in adjacent seats. Two of them, Amina and Unai, want to sit together to share popcorn. In how many ways can the five friends be seated while always fulfilling this condition?
In the final of a swimming race at the Olympic Games, $8$ swimmers participate. If we know with certainty which swimmer won the gold medal and which swimmer finished in…
In the final of a swimming race at the Olympic Games, $8$ swimmers participate. If we know with certainty which swimmer won the gold medal and which swimmer finished in last place due to an injury, in how many different ways could the other participants finish in the remaining positions?
Permutations with repeated elements
How many different words can be formed with the letters of ANA? If you treated the two letters A as distinguishable ($A_1$ and $A_2$) you would get $3!=6$ arrangements, but half of them are indistinguishable when read: $A_1NA_2$ and $A_2NA_1$ are, in fact, the same word.
How many different words, whether meaningful or not, can be formed by rearranging all the letters of the word "MUNDO"?
How many different words, whether meaningful or not, can be formed by rearranging all the letters of the word "MUNDO"?
A ship communicates with other vessels using coloured flags hoisted vertically on a mast. If it has $3$ red flags, $2$ blue flags, and $1$ yellow flag, all of the same…
A ship communicates with other vessels using coloured flags hoisted vertically on a mast. If it has $3$ red flags, $2$ blue flags, and $1$ yellow flag, all of the same size, how many different signals can it send using all the flags at once?
A software engineer wants to form six-digit numerical codes using exactly the following digits in each code: $2, 2, 3, 3, 3, 4$. How many different codes can the system…
A software engineer wants to form six-digit numerical codes using exactly the following digits in each code: $2, 2, 3, 3, 3, 4$. How many different codes can the system program?
Circular permutations and paths on a grid
In a row, the first and last places are different. At a round table there is no "first seat": only the relative order matters. Rotating the whole group by one seat does not count as a new arrangement.
Another situation with the same kind of count, although it is not circular: a delivery driver who can only move right or up on a grid. Each shortest path is a sequence of "right" (R) and "up" (U) steps, and counting paths is equivalent to arranging that sequence of letters with repeats.
A family of $6$ members sits down to dinner around a circular table in a restaurant. In how many different ways can they be arranged if only their relative positions to…
A family of $6$ members sits down to dinner around a circular table in a restaurant. In how many different ways can they be arranged if only their relative positions to one another matter and not the exact chair they occupy in the room?
On the map of a city with streets laid out in a grid, a bicycle courier must travel from point $A$ to point $B$, moving only to the right and upwards. How many different…
On the map of a city with streets laid out in a grid, a bicycle courier must travel from point $A$ to point $B$, moving only to the right and upwards. How many different paths of minimum length can they take to make the delivery?