Permutations and Combinations 2 Question 23
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
$$ \frac{m !(m+1) !}{(m-n+1) !} $$
(1983, 2M)
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Answer:
Correct Answer: 23. $(n=9$ and $r=3)$
Solution:
- Since, $m$ men and $n$ women are to be seated in a row so that no two women sit together. This could be shown as
$$ \times M _1 \times M _2 \times M _3 \times \ldots \times M _m \times $$
which shows there are $(m+1)$ places for $n$ women.
$\therefore$ Number of ways in which they can be arranged
$$ \begin{aligned} & =(m) !{ }^{m+1} P _n \\ & =\frac{(m) ! \cdot(m+1) !}{(m+1-n) !} \end{aligned} $$
