7 boys be seated at a round table. $X$ is the number of ways in which two particular boys are next to each other, $Y$ is number of ways in which they are separated. Then  $X+Y$ is 

7 boys can be seated around a table in 6! Ways consider two particulars boys to be one. 

Hence total number of boys $= 6$   

Now 6 boys can be seated around a table in $5!$ Ways and two Particular boys can be arranged among themselves in $2!$ Ways.  

Hence total number of ways  $=5! 2! =240$

Total number of ways in which particular boys are separated $=6! – 5! 2! =240 = 480$
  $\therefore X+Y = 720$