The number of ways in which $m+n(n\le m+1)$ different things can be arranged in a row such that no two of the n things may be together, is

First deduct then things and arrange the m things in a row taken all at a time, which can be done in rn! ways. Now in$(m + 1)$ spaces between them (including the beginning and end) put the n things one in each space in all possible ways. 

This can be done in ${}^{m+1}P_n$ ways.

So, the required number $=m{!}^{m+1}{P}_n=\frac{m!(m+1)!}{(m+1-n)!}$