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Question
mmen and n women are to be seated in a row so that no two women sit together. If m > n, then the number of ways in which they can be seated is
mmen and n women are to be seated in a row so that no two women sit together. If m > n, then the number of ways in which they can be seated is
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Solution
The correct option is A
First arrange m men, in a row in m! ways. Since n< m and no two women can sit together, in any one of the m! arrangement, there are (m+1) places in which n women can be arranged in m+1Pn, ways. ∴ By the fundamental theorem, the required number of arrangements of m men and n women (n<m) = m!.m+1Pn = m!.(m+1)!{(m+1)−n}! = m!(m+1)!(m−n+1)!
First arrange m men, in a row in m! ways. Since n< m and no two women can sit together, in any one of the m! arrangement, there are (m+1) places in which n women can be arranged in m+1Pn, ways. ∴ By the fundamental theorem, the required number of arrangements of m men and n women (n<m) = m!.m+1Pn = m!.(m+1)!{(m+1)−n}! = m!(m+1)!(m−n+1)!