Question

The number of ways of arranging 8 men and 4 women around a circular table such that no two women can sit together, is
a)8! b.) 4! c.) 8! 4! d.) 7! 8p4

Answer 1

Hint:- Here the concept of permutation and combination is required to solve this question. First we will use permutations to arrange 8 men in a circular table after that use the concept of combination to arrange 4 women.
Complete step-by-step answer:
The above question is of the topic permutation and combination. In the question mentioned above, we have to arrange 8 men and 4 women around a circular table such that no two women can sit together.
So, the number of ways that no two women sit together is such that to first arrange all the men in a circle and then place the women in places between them.

Here, ’M’ denotes the men on the circular table
So, the number of ways of arranging 8 men in a circle is $\left( {n - 1} \right)\;!\, = \left( {8 - 1} \right)\;!\, = 7!$
And now, 4 women are present ; so number of ways of placing 4 women in 8 places between men are
     8 p4 $\left[ {^n{{\text{P}}_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}} \right]$
$\therefore $ Total number of ways = 8 p4 x 7 !
               $ = \boxed{7!8{{\text{p}}_4}}$

Note:- Remember the formula of permutation and circular combination. The value of $^n{{\text{P}}_r} = \dfrac{{n!}}{{\left( {n - r} \right)!}}\;:$ where ‘r’ is the number of different combinations, and ‘n’ is the set of objects.