Q46 / 110 · VITEEE 2022 · Mathematics
If \(\mathrm{m}\) men and \(\mathrm{n}\) 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
- A\(\frac{\mathrm{m} ! \mathrm{n} !}{(\mathrm{m}-\mathrm{n}+1) !}\)
- B\(\frac{(\mathrm{m}+1) !(\mathrm{n}+1) !}{(\mathrm{m}-\mathrm{n}+1) !}\)
- Correct answerC\(\frac{\mathrm{m} !(\mathrm{m}+1) !}{(\mathrm{m}-\mathrm{n}+1) !}\)
- DNone of these
Solution
The \(\mathrm{m}\) men can be seated in \(\mathrm{m}\) ! ways When they are seated, there are \((\mathrm{m}+1)\) places, shown by \(\times\) where \(\mathrm{n}\) women can sit. Then no two women would be together as shown below \(\times \mathrm{M} \times \mathrm{M} \times \mathrm{M} \times \ldots \ldots \times \mathrm{M} \times\)
Then \(\mathrm{n}\) women can arrange themselves in these \((\mathrm{m}+1)\) places in \({ }^{\mathrm{m}+1} \mathrm{P}_{\mathrm{n}}\) ways.
Hence the required no. of ways
\(=m ! \times{ }^{m+1} P_n=\frac{m !(m+1) !}{(m-n+1) !}\)
Hence, the correct answer is option: 'C'.