In how many ways can one blue ball, one yellow ball, three green balls, two white balls, one purple ball and one violet ball be arranged (without missing any ball in the arrangement) in a line if the same coloured balls are indistinguishable among themselves?

We have 1 blue,1 yellow, 3 green, 2 white, 1purple and 1 violet ball. That makes a total of 9 balls which can be arranged in 9! ways but since 3 green and 2 white balls are indistinguishable and themselves can be arranged in 3! and 2! ways respectively, so we will have to divide them with total to negate repetitions.

Therefore, total number of ways = (9!)/(3!2!) = 30240.