Probability of a Two Boxes Having The Same Number of Distinct Balls

Given counts of k colored balls summing to 2n, compute the probability that after a uniform random shuffle and splitting the first n balls into box A and the rest into box B (boxes labeled), both boxes contain the same number of distinct colors. Core challenge is combinatorial: enumerate all ways to allocate per-color counts to box A summing to n and use multinomial coefficients to count arrangements and compare distinct-color counts to accumulate favorable vs total outcomes.