Marshal the Line-Arranger

Marshal had felt the magic of order since he was knee-high. His family ran processions in their home town — festival lines, wedding marches, lantern walks — and young Marshal's job was to decide who went where. One evening, fretting over a line of three lantern-bearers, he discovered something that changed how he saw everything.

"For the very first spot," his aunt asked him, "how many bearers could you choose?" "Three. Any of them." "And once that one's placed, for the second spot?" "Only two left to pick from." "And the last spot?" "Just the one who's left." "So multiply them," his aunt said. "Three choices, then two, then one. Three times two times one is six. Six different orders for three bearers — six different processions hiding in the very same three people." Marshal had counted them out, astonished, and found exactly six. Not five, not seven. Six. The multiplying worked. From then on he saw arrangements everywhere, each one its own small world, and he learned the trick that counted them: start with all your choices, then one fewer, then one fewer still.

When he was grown, Marshal marched the long road to DiscreteQuest, because he'd heard it honored the things most folks overlooked. The head of the academy was an old owl with a slow, considering voice. "What is a permutation?"

Marshal stood very straight. "A permutation is an arrangement where the order counts. Same things, lined up differently, and every different line-up is a different answer. To count them, you multiply your choices down — the first spot has the most, then one fewer for the next, and so on, to the last." He spread his wings. "Three things make six arrangements. Four things make twenty-four. Five things make a hundred and twenty. They pile up fast, because every new thing multiplies in." "And when the order does not matter?" the owl asked gently. "Then it's a different question, and not mine to answer. I'm the order fellow. Swap two of my things and I've got something new every time. That's my whole craft." The owl nodded, slow and pleased. "You are appointed."

Marshal's favourite thing was watching a worried student discover how arrangements blossom. A nervous mole named Pim came to him clutching a photo frame. "I have to seat three of my friends in a row for the class photo, and they each want the 'best' spot, and I don't know how many ways there even are, and I'm panicking." "Then let's not panic. Let's count," Marshal said warmly. "First seat — how many friends could sit there?" "Three." "Sit one down. Second seat — how many left to choose?" "Two." "Last seat?" "Just one." "Three, then two, then one. Multiply: six."

He laid out all six in front of Pim, swapping carefully each time. "Six photos, every one different, every one fair in its own way. You're not picking the 'one right order,' Pim. You're discovering there are six lovely ones, and you get to choose your favourite." Pim let out a long breath. "Six isn't scary. Six I can handle." "Six you can hold in your hands," Marshal agreed. "And if it were ten friends? Well — ten times nine times eight, all the way down, more than three million line-ups. But the trick never changes. Multiply your choices down."