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.
What is the probability that each child has one type of candy that they have strictly more of than every other trick-or-treater? Give your (exact!) answer in a lowest terms fraction.
November update: correct solutions to this puzzle have come in more slowly than others, so we are going to keep it up for an extra month and will have a new puzzle on the site in early December.