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Question
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\[\frac{m! (m + 1)!}{(m - n + 1) !}\]
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Solution
'm' men can be seated in a row in m! ways.
'm' men will generate (m+1) gaps that are to be filled by 'n' women = Number of arrangements of (m+1) gaps, taken 'n' at a time = m+1Pn = \[\frac{\left( m + 1 \right)!}{\left( m + 1 - n \right)!}\]
∴ By fundamental principle of counting, total number of ways in which they can be arranged =\[\frac{m!\left( m + 1 \right)!}{\left( m - n + 1 \right)!}\]
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