Q. 105.0( 1 Vote )

m men and n women are to be seated in a row so that no two women sit together. If m>n then show that the number of ways in which they can be seated as

Answer :

Given: there are m men and n women where m>n


To find: out their possible way of sitting arrangements in a row such that no two women sit together


Let m = 2 then possible arrangement is


_ m _ m _


Here 3(2 + 1) gaps for women can by made by 2 men so that no two of them comes together


When m men are there, m seats can be occupied by men and m + 1 seat can by women


Formula used:


Number of arrangements of n things taken all at a time = P(n, n)



Total number of arrangements of men


= the number of arrangements of m things taken all at a time


= P(m, m)




{ 0! = 1}


= m!


Formula used:


Number of arrangements of n things taken r at a time = P(n, r)



Total number of arrangements of women


= the number of arrangements of m + 1 things taken n at a time


= P(m + 1, n)




Hence, total possible arrangements of m men and n women in a row such that no two women come together




Hence, Proved.


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