Permutations
Seven runners cross the finish line of a race in Chapultepec Forest, with no ties. How many different rankings are possible? This is a question about ordering: placing all the elements of a set, one after another. This is called a permutation. Counting orderings correctly helps you calculate probabilities: possible cases and favorable cases.
Permutations of n distinct elements
For first place there are $n$ options. Once that is chosen, there are $n-1$ left for second place; $n-2$ for third; and so on until only $1$ option remains for the last place. They are multiplied because each choice combines with all the following ones (multiplication principle).
$$P_n = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1 = n!$$Factorials grow very fast, so it is best to simplify before calculating. For example, $\dfrac{9!}{7!} = \dfrac{9 \times 8 \times 7!}{7!} = 72$.
Calculate the numerical value of the following expressions involving factorials: $\frac{8!}{5! \times 3!}$ and $\frac{12!}{10!}$.
Calculate the numerical value of the following expressions involving factorials: $\frac{8!}{5! \times 3!}$ and $\frac{12!}{10!}$.
A group of $6$ friends goes to the theater 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 theater 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?
Simplify the following algebraic expression involving factorials, expressing the result in its simplest form: $\frac{(n+2)!}{n! \times (n+1)}$.
Simplify the following algebraic expression involving factorials, expressing the result in its simplest form: $\frac{(n+2)!}{n! \times (n+1)}$.
Equations with factorials
To solve equations with factorials, expand each factorial down to the smallest factor that appears, simplify, and you are left with an equation with polynomials.
Find the value of the natural number $x$ that satisfies the following equation, justifying your answer mathematically: $P_x = 30 \times P_{x-2}$.
Find the value of the natural number $x$ that satisfies the following equation, justifying your answer mathematically: $P_x = 30 \times P_{x-2}$.
Simplify the algebraic fractions and solve the following equation containing factorials, finding the value of $x$: $\frac{(x+1)!}{(x-1)!} - \frac{x!}{(x-2)!} = 14$.
Simplify the algebraic fractions and solve the following equation containing factorials, finding the value of $x$: $\frac{(x+1)!}{(x-1)!} - \frac{x!}{(x-2)!} = 14$.
Permutations with position restrictions
Sometimes part of the ordering is already fixed: two people who must sit together, an element that must go first or last. You do not apply $n!$ directly: first you reason out the restriction and then you order what is left free.
To alternate, such as 3 boys and 3 girls in a line, each pattern is counted separately: the boy-girl-boy... pattern has $3!$ ways to place the boys and $3!$ to place the girls. The pattern starting with a girl is just as numerous, so you multiply by $2$.
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?
A group consisting of $3$ boys and $3$ girls are going to sit in a row of $6$ seats to watch a concert. In how many ways can they do so if they must sit such that the…
A group consisting of $3$ boys and $3$ girls are going to sit in a row of $6$ seats to watch a concert. In how many ways can they do so if they must sit such that the genders strictly alternate (boy-girl-boy... or girl-boy-girl...)?
On a school library shelf, $4$ different Mathematics books, $3$ different Physics books, and $2$ different Chemistry books are to be placed in a row. If the librarian…
On a school library shelf, $4$ different Mathematics books, $3$ different Physics books, and $2$ different Chemistry books are to be placed in a row. If the librarian requires that books on the same subject must always stay together as a block, in how many different ways can they be arranged on the shelf?
Permutations with repeated elements
How many different words can be formed with the letters of ANA? If the two A's were distinguishable ($A_1$ and $A_2$) there would be $3! = 6$ orderings, but $A_1NA_2$ and $A_2NA_1$ read the same: they are the same word.
With a restriction on the ends, you fix them first and permute the rest. For ESTADISTICA (11 letters) starting with E and ending with A, you remove those two letters and 9 remain: S, T, A, D, I, S, T, I, C. S ($2$), T ($2$) and I ($2$) are repeated: $\dfrac{9!}{2!\,2!\,2!}$.
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"?
To configure a secure password, an application requires using exactly the characters $A, B, C, 1, 2$, without repeating any of them. How many different passwords can be…
To configure a secure password, an application requires using exactly the characters $A, B, C, 1, 2$, without repeating any of them. How many different passwords can be created in the system?
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?
How many different words, whether meaningful or not, can be formed using all the letters of the word "CAMPANA"?
How many different words, whether meaningful or not, can be formed using all the letters of the word "CAMPANA"?
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?
How many different words, meaningful or not, can be formed by rearranging all the letters of the word "ESTADISTICA" so that they must begin with the letter $E$ and end…
How many different words, meaningful or not, can be formed by rearranging all the letters of the word "ESTADISTICA" so that they must begin with the letter $E$ and end with the letter $A$?
Circular permutations and grid paths
In a line, 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 one seat does not count as a new arrangement.
Another situation with the same kind of counting: a delivery person on a bicycle who can only go right or up along the streets of a grid. Each shortest path is a sequence of "right" (R) and "up" (U) steps, and counting paths is ordering a word with repeated letters.
If a node is blocked, you count all the paths and subtract those that pass through it (the ones that go from A to the node, multiplied by the ones that go from the node to B).
Back to the beginning: the 7 runners can be ordered in $7! = 5{,}040$ ways; if you only wanted to guess the complete ranking at random, your probability would be $\frac{1}{5{,}040}$.
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?
Eight diplomats meet at an international summit and must sit around a large round table. Two of the representatives are strongly hostile to each other, and protocol…
Eight diplomats meet at an international summit and must sit around a large round table. Two of the representatives are strongly hostile to each other, and protocol requires that they do not sit next to each other. In how many different ways can they be seated at the table fulfilling this restriction?
An autonomous robot moves through a logistics grid from node $A$ to node $B$, moving exclusively to the right and downwards. Due to floor repairs, node $C$ is blocked…
An autonomous robot moves through a logistics grid from node $A$ to node $B$, moving exclusively to the right and downwards. Due to floor repairs, node $C$ is blocked and the robot cannot pass through it. How many valid paths exist to complete the route?
Order in a list and sum of numbers
Two classic problems use permutations with care.
Position in a dictionary. With the letters of LIBRO in alphabetical order (B, I, L, O, R), count the words that come before: those starting with B ($4! = 24$) and I ($24$), before those starting with L. That makes $48$ so far. Among those starting with L, the next letters B, I, O, R: those starting with LB are $3! = 6$, and those starting with LI and then B: LIB comes before LIO and LIR. The first one with LIB is LIBOR and then LIBRO, so there are $48 + 6 + 1 = 55$ words before it; LIBRO is number $56$.
Sum of numbers. With the digits $1, 2, 3, 4$ there are $4! = 24$ four-digit numbers with distinct digits. In each position, each digit appears $3! = 6$ times. The sum of the digits in one position is $6 \times (1+2+3+4) = 60$. Multiply by $1{,}111$ (units, tens, hundreds and thousands): $60 \times 1{,}111 = 66{,}660$.
Calculate the total sum of all $4$-digit integers with distinct digits that can be formed using exclusively the digits $1, 2, 3$ and $4$.
Calculate the total sum of all $4$-digit integers with distinct digits that can be formed using exclusively the digits $1, 2, 3$ and $4$.
If all possible words, meaningful or not, are generated by permuting the letters of the word "LIBRO" and then strictly ordered alphabetically (like in a dictionary),…
If all possible words, meaningful or not, are generated by permuting the letters of the word "LIBRO" and then strictly ordered alphabetically (like in a dictionary), what exact position will the word "LIBRO" occupy in that list?