m men and w women are to be seated in a row so that no two women sit together. If  m > w, then the number of ways in which they can be seated is:

We first arrange the m men. This can be done in m! ways. After m men have taken their seats, the women must choose w seats out of  (m + 1) seats marked with X  below.XMXMXMX…XMX 1st 2nd 3rd m th (m+1) th They can choose w seats in  m+1Cw ways and arrange w women in w! ways.Thus, the required number of arrangements ism! m+1Cw(w!)=m!(m+1)!w!w!(m+1−w)!=m!(m+1)!(m+1−w)!