m men and n women are to be seated in a row so that no two women sit together

if mn the show that the number of ways in which they can be seated as

m!(m+1)! / (m-n+1)!

Let the men take their seats first. They can be seated in ^mP_m ways as shown in the fig.

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)P_n ways

and Hence, the total no. of ways so that no women sit together is

(^mP_m) x (^m+1P_n) = m!(m+1)!/ (m-n+1)!