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
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.