for getting at least one ball of every color in worst case you have to get all 8 green balls, and 7 white balls and after that you can pick just one ball from the bag after all these drawn then that ball will be definitely of red color, So, in this way you satisfy the condition for at least one ball of all color.
so 8 + 7 + 1 = 16,
and this is the minimum no of balls in order to be 100% sure that picked up balls contain all three colors.
if you pick less than 16 then it is not guaranteed that you will satisfy the condition, suppose you picked15 balls there may be a chance of getting 8 green and +7 white ball, and in this combination, you don't have a white ball which dis-satisfying the condition.
hence minimum is 16 only.