Candy Collectors II

Five children went trick-or-treating together and decided to randomly split their candy haul at the end of the night. As it turned out, they got a total of 25 pieces of candy, 5 copies each of 5 different types (they live in a small town). They distribute the candies by choosing an ordering of the 25 uniformly at random from all shufflings, and then giving the first 5 to the first child, the second 5 to the second, and so on.

Say a child dominates a candy type if they hold strictly more pieces of that type than each of the other four children.

What is the probability that every child dominates at least one type of candy?

Answer format: an exact fraction in lowest terms.