Check how many ways you can distribute the balls between the boxes.

Consider that one way you will use (x1, x2, x3, ..., xk) where xi is the number of balls from colour i. The probability of achieving this way randomly is ( (ball1 C x1) * (ball2 C x2) * (ball3 C x3) * ... * (ballk C xk)) / (2n C n).

The probability of a draw is the sigma of probabilities of different ways to achieve draw.

Can you use Dynamic programming to solve this problem in a better complexity ?