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Question
Suppose m men and n women are to be seated in a row so that no two women sit together. If m > n, show that the number of ways in which they can be seated is `(m!(m + 1)!)/((m - n + 1)1)`
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Solution
Let the men take their seats first.
They can be seated in mPm ways as shown in the following figure
From the above figure, we observe, that there are (m + 1) places for n women.
It is given that m > n and no two women can sit together.
Therefore, n women can take their seats (m+1)Pn ways
And hence the total number of ways so that no two women sit together is
`(""^nP_m) xx (""^(m + 1)P_n) = (m!(m + 1)!)/((m - n + 1)1)`
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