Question

$m$ men 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?
A. $\dfrac{{m!(m + 1)!}}{{(m - n + 1)!}}$
B. $\dfrac{{{\text{m}}!({\text{m}} - 1)!}}{{({\text{m}} - {\text{n}} + 1)!}}$
C. $\dfrac{{({\text{m}} - 1)({\text{m}} + 1)!}}{{({\text{m}} - {\text{n}} + 1)!}}$
D. None of these

Answer 1

Hint: When a question is based on permutation and combination, we know that ‘r’ objects can be arranged from a collection of ‘n’ objects in ${}^n{P_r}$ ways. According to the question first, we will find the number of total possible ways for $m$ men and then the ways for $n$ women arrangement is evaluated to get the solution of the given problem.

Formula Used:
The formula used in the problem to find out the number of ways of arrangements from permutation is $^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$

Complete step by step Solution:
Let $m$ men are to be arranged in $m$ ways and $n$ women are to be arranged in $n$ ways.
The total number of ways in which the $m$ men can be arranged is ${}^m{P_m}$.
Now according to the question no two women are to be seated together, for this, we have $m + 1$ places for women. Therefore, the total number of ways in which $n$ women can be seated is ${}^{m + 1}{P_n}$
Now, the total number of ways in which women and men can be seated will be: -
$ = {}^m{P_m} \times {}^{m + 1}{P_n}$
Also, we know the formula to find out the number of ways of arrangements from permutation is $^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$
Applying this formula, we get the total number of ways in which both men and women can be seated as: -
$ = \left( {m!} \right) \cdot \dfrac{{(m + 1)!}}{{(m - n + 1)!}}$
$ = \dfrac{{m!(m + 1)!}}{{(m - n + 1)!}}$

Hence, the correct option is (A).

Note: First we have to determine the number of ways in which $m$ men can be seated after that number of ways in which $n$ women can be seated must be calculated, then use the required formula $^n{P_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}$ to determine the total number of ways in which both men and women can be seated together.