There are 2 identical white balls, 3 identical red balls and 4 green balls of different shades. The number of ways in which they can be arranged in a row so that atleast one ball is separated from the balls of the same colour, is :

Explanation

Number of ways balls are arranged in a row so that at least one ball is separated from the balls of same colors,

= Total Number of ways - All identical balls together

Total number of ways balls can be arranged `n(S) = (9!)/(2!3!)`

`n(S) = (9!)/12 = (9**8**7!)/12 = 6 (7!)`

The number of ways balls are arranged so that identical balls are together `n(T) = 3!4!`

So, the number of ways balls are arranged so that at least one ball is separated ` =
n(S) - n(T) = 6 (7!) -3!4! = 6 (7!) -64! = 6(7!-4!)`