# Tagged Questions

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### Permutation with atleast n unique characters

I came across this question on Google APAC 2015. I am slightly weak with permutations. The problem goes like this: There is a password. We know the length of the password and the characters used ...
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### Probability to find Divisibility of an integer by prime numbers [closed]

What is the probability that a positive integer n<100 is divisible by a prime number p<100?
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### Numbers which are writable as a sum of permutation pairs

We say that $N$ is writable as a sum of permutation pair $\{a,b\}$ if $a+b=N$, $a\neq b$ and $a$ and $b$ are permutations of each other (e.g. $321 = 156 + 165 = 147 + 174 = ...$). Looking at 3-digit ...
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### Count ways to sit men women in row of size K

Suppose we are given N men and M women.They are to sit in a row of size K such that no two women sit next to each other.What are the number of ways. Like if suppose their are 3 men and 2 women and ...
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### How to check if $\text{position}\left(\frac{a + b} 2\right )$ is in range $\text{position}\left(a\right )$ and $\text{position}\left(b\right )$

Given a permutation of $n$ number $1, 2, 3,\dots,n$. How to check if it is exist $a,\ b$ with the same parity such that $\frac {a + b} 2$ is between $a, b$. How to solve this problem efficiently ? ...
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### Partition numbers with restriction on the greatest part *and* on the number of positive parts

I’m looking at partition numbers. OEIS A008284 says that the number of partitions of $n$ in which the greatest part is $k$, $1 \le k \le n$, is equal to the number of partitions of $n$ into $k$ ...
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### Count swap permutations

Given an array A = [1, 2, 3, ..., n]: ...
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### How find this $aA_{m+1}=\overline{\sigma_{0}\sigma_{1}\sigma_{2}\cdots\sigma_{m}}$

Question let $m$ is positive numbers,and such $m\ge 5$,and $$A_{m+1}=\overline{1234\cdots m}=1\times (m+1)^{m-1}+2\times (m+1)^{m-2}+\cdots+(m-1)\times (m+1)+m$$(or see ...
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### Expected Value of this function

Let’s consider a random permutation p1, p2, …, pN of numbers 1, 2, …, N and Function F is calculated as F=(X[2]+…+X[N-1])^K, where ...
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### What is the maximum number of iterations before a sequence is repeated

$A = \{a,b,c,d,e\}$ $B = \{f,g,h\}$ $C = \{i,j\}$ $D = \{0,1,2,3,4,5,6\}$ Suppose a four-tuple is constructed by extracting one element from each set at each successive iteration. The stipulation ...
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### Describing the pattern in which iterations make two, cyclic sets equal

$A = \{a,b,c,d,e\}$ $B = \{a,b,c\}$ $C = \{0,1,2,3,4,5,6\}$ The first few iterations are as follows: $1.$ $a,a,0$ $2.$ $b,b,1$ $3.$ $c,c,2$ $4.$ $d,a,4$ $5.$ $e,b,5$ $...$ I'm trying to ...
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### Partitions of n with certain conditions

Let $p$ be prime and $n$ be any integer. Suppose $t=(n^{a_n}, \dots, 2^{a_2}, 1^{a_1}) \vdash n$, (i.e. $t$ is a partition of $n$, where we group repeated integers, so, for example, $2^{a_2}$ means ...
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$N$ students are to be provided with gifts We know that the $i$'th student wants to get at least $a_i$ gifts. The teacher wants to give distinct gifts meaning if he give $x$ gifts to one student then ...
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### Numbers permutation

Given $n$ numbers and $k$ positions I want the total number of permutations of these n numbers on these $k$ positions if repetition is allowed and if the following two arrangements are considered ...
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### Number of ways of arranging numbers with given max difference

How many ways are the there to arrange n numbers out of m numbers (1 to m) so that the difference between the max and min numbers of those n numbers is D which is given. For example : n = 4 m = 3 ...
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### Permutations with some fixed numbers

You have to fill 4 spaces with 3 numbers (4, 5, 6) such that the numbers 4 and 6 appear atleast once in every case. Find the number of such unique permutations. [Ans. 50] How do you go about solving ...
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### Minimum moves to transform a list to another?

Given two list of n positive elements. We are allowed to perform only one transformation which is to increment each element of the list except one. What are the minimum number of transformation ...
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### Number of ordered pairs $(x,y)$ with $x,y \in \{ 1, \dots, N\}$ such that sum of numbers in pairs is divisible by $k$?

Given $N$ and $k$. Find the number of ordered pairs $(x,y)$, with $x,y \in \{ 1, \dots, N\}$ such that sum $x+y$ is divisible by $k$. and x less than y. example $N=10, k=4$ answer = $10$ Pairs- ...
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### Path through multiplication table mod p

Suppose I have a multiplication table mod $p$, with the $0$ row and column excluded. Suppose I wish to circle a different number from $1$ through $p-1$ in each column so that no two numbers are ...
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### Find the sum of all 4-digit numbers formed by using digits 0, 2, 3, 5 and 8?

Find the sum of all 4-digit numbers formed by using digits 0, 2, 3, 5 and 8. Finding the total number of number is possible, but how can the sum be found?
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### Factorial related problems

How many zeros are there in $25!$ ? My answer was $6$. But i solved it by finding how many numbers are divisible by $5$ and $2$.here i was told to find out the zeros at the last end. But what is the ...
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### circle eliminator theory

what is the formula for that, some peoples are standing in a circle remove every second person that in the end only single person remains for example if 5 persons are in circle than first 2nd and ...
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### Listing the elements of $A(3)$

List the elements of $A(3)$ and give the order of each of them. This is about permutations in number theory ... to clarify that $A(n)$ Thanks!
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### Finding the sign of each permutations

How to find the sign of each of the following permutations? 1, (1 2 3 4 5)(8 7 6)(10 11) 2, (1 3 5 7 9 11)(2 4 6 8 10) 3, (1 2)(3 4)(5 6 7 8)(9 10) 4, (1 2 3 4 5 6 7 8)(1 8 7 6 5 4 3 2) Help ...
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### Consider $P(n)$ as a number of $n$-permutations, which each cycle have even length, and … [duplicate]

Consider $P(n)$ as a number of $n$-permutations, which each cycle have even length, and $N(n)$ as a number of $n$-permutations, which each cycle have odd length. Calculate $P(2n)-N(2n)$
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### Six Unique numbers which generate unique sum during addition

I have a Six numbers which are like points that satisfy certain condition. If the condition is satisfied that point will be given or else 0 will be assigned as points. I am Storing the points in ...
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### Need help in determining where this pascal's triangle-like sequence comes from.

I have a very interesting problem in that a program that I am running has generated a sequence of numbers that act like the pascal's triangle but have somehow built more structure into it. I have been ...
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### How to find the last non-zero digit in ${^n\!P_k}$?

What is the procedure of finding the last non-zero element in ${^n\!P_k}$?
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### Jacobi symbol and invertibility of $m$ for an odd $n$

I have asked a similar question here before, and received a nice answer. I think that the next question here is equivalent, but can't seem to be able to prove it. Here goes: Given an odd $n$, I want ...
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### How to arranged two or more different colored blocks in all possible ways?

Is any algorythem that can arrange two or more different blocks in all possible ways.. in series (rows and columns.)? If I have two colored(red and blue) blocks and I try to arranged in one possible ...
This is a completely random question that just happened to come to mind recently and I was wondering if the MathSE community had anything to say about it. Let $n > 1,b > 1$ be integers and ...
Is it possible to efficiently factor a semiprime given a bit-permutation relating the factors? For example, suppose we have $n = p * q = 167653$; in this case, $p = 359 = 101100111_2$ and \$q = 467 = ...