A New Version of Menages Problem

The problème des ménages (married couples problem) introduced by E.Lucas in 1891 is a classical problem that asks the number of ways to arrange n married couples around a circular table, so that husbands and wives are in alternate places but no couple is seated together. In this paper we present a new version of Menage Problem that carries the constraints consistent with the Muslim culture. The following problem, introduced by Lucas in 1891, is known as the problème des ménages: in how many ways can seat n couples at a circular table so that men and women are in alternate places and no husband will sit on either side of his wife? (see [1], [2]). In this paper we consider the following problem: Suppose there are nk k -tuples (one husband and k − 1 wife/wives), 2 ≤ k ≤ r, m single men and w single women. In how many ways we can seat them around a circular table such that: 1. All members of a family sit together; 2. No man sits adjacent to a woman except his own wife/wives; 3. A husband having more than one wife will be surrounded by his wives. For a real situation we restrict ourselves to r = 5, and suppose that there are: n2 couples (husband and wife), n3 triples (husband and two wives), n4 quadruples (husband and three wives), n5 pentuples (husband and four wives), m single men and w single women. 2 Ahmad Mahmood Qureshi Theorem 1.1. The solution of this problem for n = 5 and n2 even is 2345 · 345 · n2! ( 2 2 )! · ( 2 2 +m−1)! ( 2 2 −1)! · (2 2 +n3 +n4 +n5 +w− 1)! Proof: Let us code a man by 1 and a woman by 0. We shall first find the number of ways to arrange the married persons around the circular table. Since all members of a family have to sit together, the possible coding arrangements of the given tuples (without distinguishing the wives) could be: Couple 01 10 Triple 001 010 100 Quadruple 0001 0010 010