Question

in how many ways 5 white balls and 3 red balls be arranged in a row sothat no 2 red balls are together ?

Answer 1

1 WAY IS THAT KEEP 1 W BALL THEN R BALL & SO ONE
2 WAY IS TO KEEP THE R BALLS AT THE TWO ENDS AND IN B/W  W BALLS
3,4,5,6 WAY IS BY KEEPING 1 R BALL AT THE END AND CHANGING THE OTHER R BALL'S POSITION IN SUCH A WAY THAT ATLEAST 1 W BALL IS IN B/W

Answer 2

        1    W    2     W    3     W   4     W    5    W    6

Let the white balls W be arranged as above.  There are 6 slots in which red balls can be placed.    There are 3 red balls to place in any 3 of 6 slots.

So  the number of ways:  ${}^6C_3$  = 6 ! / (3! * 3!)
             = 20 ways.