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N people are invited to a dinner party and they are sitting on a round table. Each person is sitting on a chair there are exactly N chairs. So each person has exactly two neighboring chairs, one on the left and the other on the right. The host decides to shuffle the sitting arrangements. A...

How many pairs(A,B) are there up to n such that GCD(A,B)=B without those pairs(A,B), where B^2=A.If we consider 5 as n we have 25 possible pairs.They are (1,1),(1,2),(1,3),(1,4),(1,5),(2,1),(2,2),(2,3),(2,4),(2,5), ................. and (5,5). Of them, 8 pairs satisfy the above condition.My...

There are n distinct points in the plane, given by their integer coordinates. Find the number of parallelograms whose vertices lie on these points. In other words, find the number of 4-element subsets of these points that can be written as {A, B, C, D\} such that AB||CD,and BC||AD. No four...

A Linear Diophantine Equation is of the following form: Ax+By+C=0, where x1≤x≤x2 and y1≤y≤y2. If the value of A,B,C,x1,x2,y1,y2 are given and x1≤x2 and y1≤y2, then how many solutions can be found? How can I find out the total number of solutions according to the above condition. I asked the...

How many pairs(A,B) are there up to n such that GCD(A,B)=B without those pairs(A,B), where B^2=A. If we consider 5 as n we have 25 possible pairs.They are (1,1),(1,2),(1,3),(1,4),(1,5),(2,1),(2,2),(2,3),(2,4),(2,5),...,(5,5). Of them, 8 pairs satisfy the above condition.My main objective is to...

How many pairs of integers (A,B) are there in the range [1,…,N], such that gcd(A,B)=B? Here, N<=10^9. I want to know the best efficient method to find out the number of pairs with proof .

How many arrangements of first N natural numbers are found ,where in first M positions; exactly K numbers are in their initial position.For example,N=5,M=3,K=2 we have 12 such arrangements.Please answer the question with a clear explanation.

I read many articles on the proof of derangement formula (based on Inclusion-Exclusion) but I couldn't understand this.Please help me. The formula is here: http://upload.wikimedia.org/math/1/9/2/1922f4c5da0db857bad09d1b352e7088.png I only know the Inclusion-Exclusion based proof with an example

Given n different objects,they have to be arranged in such a way that k objects won't occupy their initial position.How to solve this problem? For example,if n=7,k=3 how many permutations will we get so that exactly 3 objects won't occupy their initial position?