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Questions tagged [permutations]

For questions related to permutations, which can be viewed as re-ordering a collection of objects.

The word permutation has several possible meanings, based on context. In combinatorics, a permutation is generally taken to be a sequence containing every element from a finite set exactly once. Permutations of a finite set can be thought of as exactly the ways in which the elements of the set can be ordered.

In group theory, a permutation of a (not necessarily finite) set $S$ is a bijection $\sigma : S \to S$. The set of all permutations of $S$ forms a group under composition, called the symmetric group on $S$.

Reference: Permutation.

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how many 5 digit numbers are there with distinct digits?

I found this question on gre forum, it's answer was given by this expression: $9\cdot9\cdot8\cdot7\cdot6$ which I heard in school as well. What I tried to do was: for numbers from index $4$ to index $1$, we can use any of the $10$ numbers $(0-9)$…
Max
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Permutations without repeats

Consider this matrix: A B C D E D C A E B B A E C D E D B A C C E D B A The letters are arranged so that no row and no coulumn contains the same letter twice (Sudoku style). Let's call the number of diffent letters $n$ (5 in the above example).…
oz1cz
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$n\times n$ matrices whose entries are each 1 or -1 be such that the sum of the entries in each row and in each column is 0

Find the number of $n\times n$ matrices whose entries are each 1 or -1 be such that the sum of the entries in each row and in each column is 0. ($n$ is given even) I attempted this question in a way similar to an existing similar question about…
Gaurang Tandon
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Simplify these products of permutations

Can someone please explain how to get these answers? Everytime I think I understand the method, I end up getting a completely different answer to the one provided. How exactly do you go about simplifying these? [Note: Read from left to…
Mathlete
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Airport queues permutation

$k$ people get into a plane and walk into a hall where they are assigned to at most $n$ queues. The number of ways in which this can be done is ? The answer I was given is $n(n+1)...(n+k-1)$. I tried several times to make sense out of it, but to…
Meera Unni
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Solving Permutation

There are 5 positions on a flagstaff and 4 different colors of flags ( at least 5 of each color). How many different signals can be made by displaying 5 flags simultaneously?
Madisson
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Prove that every permutation is the product of disjoint cycles

I need to prove that every permutation $\sigma \in S_n, \sigma \neq id$ can be written as a distinct product of disjoint cycles and don't really know where to start. The statement is totally clear, but I cannot think of an elegant way to proof its…
user7802048
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A silly confusion with permutations?

I'm reading Beardon's Algebra and Geometry, I'm a little confused about this exercise: If $n$ is not divisible by $3$, then $\rho$ fixes some $k$. Suppose $n=5$, then we have the set $\{1,2,3,4,5\}$. As $\rho$ is a $3-$ cycle, we write it as -…
Red Banana
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Find the number of restricted permutations

Find the number of permutations of ABCDEFGHI such that the patterns ABC, DEF, GHI do not occur. I am getting $181434$. $$(9!-9!/3!\cdot3)-3!$$
Subhasis Biswas
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While solving for $nP5 = 42 \cdot nP3$, $n > 4$

...if we cancel out $n!$ on both sides we get to a complex quadratic which gives a wrong result. But, if we cancel out the $(n-5)!$ and $(n-3)!$ on their respective sides of the equation and then solve the quadratic and use the constraint $n>4$ we…
Divesh M
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How many 3 digit numbers can be formed using the digits 0,1,2,3,4 if 2 and 3 are always included?

My attempt : Fixing $2$ and $3$ at the tens and units place. Hundreds place can be filled up by $(1,4)$. Hence $2$ numbers can be formed in this case. Fixing $2$ and $3$ at the hundreds and tens place. Units place can be filled up by $(0,1,4)$.…
John
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Writing Permutations as composition of transpositions

I am currently learning about, and I am also going to give a short presentation on, a theorem that states the following: Theorem: The number of transpositions whose product is a given permutation of a finite set is either always even or always…
Smeef
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Number of squares in a rectangle

Given a rectangle of length a and width b (as shown in the figure), how many different squares of edge greater than 1 can be formed using the cells inside. For example, if a = 2, b = 2, then the number of such squares is just 1.
g4ur4v
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Number of possible matrices according to given rule?

You want to create a matrix of size MxN using only 0 & 1. But there is rule- There must not be any submatrix with its (rows>1 & columns>1) and filled with same number. For example- for a 3x3 matrix- 0 1 1 1 0 0 1 1 0 is allowed but- 1 0 0 1…
user1437473
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Using digits $0,1,2,3,4$ and $5$ only once, how many $5$ digit numbers can be formed which are divisible by $4$?

Using digits $0,1,2,3,4$ and $5$ only once, how many $5$ digit numbers can be formed which are divisible by $4$? I worked on this problem and I have the solution as: The number of 5 digit numbers formed will be $5 \cdot 5 \cdot 4 \cdot 3 \cdot 2$,…
Mary Lopez
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