JEE Advance - Mathematics (1983 - No. 23)
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 is $$\,{{m!(m + 1)!} \over {(m - n + 1)!}}$$
Arrange the men in m! ways, then choose n spaces from m+1 spaces and arrange women in n! ways, giving m! * (m+1)Cn * n! = m! * (m+1)! / (m-n+1)!
Arrange the women in n! ways, then choose m spaces from n+1 spaces and arrange men in m! ways, giving n! * (n+1)Cm * m! = m! * (n+1)! / (n-m+1)!
Arrange men and women alternatively, so (m+n)!
Arrange men in (m-1)! ways, then arrange women in n! ways, giving (m-1)! * n!
Arrange women in (n-1)! ways, then arrange men in m! ways, giving (n-1)! * m!
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