In how many ways can 9 different colour balls be arranged in a row so that black, white, red and green balls are never together?

Let's find out the number of ways of arranging 9 different balls in a row.

That can be done in 9! = 3,62,880

Now, let's find out the number of ways to arrange nine different balls in a row so that black, white, red, and green balls come together.

Let's make them a group. And arrange all the balls in a row.

This can be done in 6!*4! = 720*24 = 17,280.

The number of ways in which 9 different colour balls can be arranged in a row so that black, white, red and green balls are never together is 

 3,62,880 - 17,280 = 345600