A quick stab at the number of possibilities for a starting Onitama outlay.

onitama-possibilities.ts

import { Set } from "typescript-collections";
const cards: string[] = [
  "Boar",
  "Cobra",
  "Crab",
  "Crane",
  "Dragon",
  "Eel",
  "Elephant",
  "Frog",
  "Goose",
  "Horse",
  "Mantis",
  "Monkey",
  "Ox",
  "Rabbit",
  "Rooster",
  "Tiger",
];

const sets = new Set<string>();

cards.forEach((card1, idx) => {
  cards.forEach((card2, idx2) => {
    cards.forEach((card3, idx3) => {
      cards.forEach((card4, idx4) => {
        cards.forEach((card5, idx5) => {
          const set = new Set<string>();
          set.add(card1);
          set.add(card2);
          set.add(card3);
          set.add(card4);
          set.add(card5);
          if (set.size() === 5) {
            sets.add(set.toArray().sort().join(","));
          }
        });
      });
    });
  });
});

console.log("Sets", sets.size());

const arrangedSets = new Set<string>();

sets.forEach((set) => {
  const arrSet = set.split(",");
  arrSet.forEach((card1) => {
    arrSet.forEach((card2) => {
      arrSet.forEach((card3) => {
        arrSet.forEach((card4) => {
          arrSet.forEach((card5) => {
            const possibleSet = new Set<string>();
            [card1, card2, card3, card4, card5].forEach((card) =>
              possibleSet.add(card)
            );
            if (possibleSet.size() !== 5) {
              return;
            }
            const arrangedSet = new Set<string>();
            const firstSet = new Set<string>();
            const secondSet = new Set<string>();
            const remainder = new Set<string>();
            firstSet.add(`a:${card1}`);
            firstSet.add(`a:${card2}`);
            secondSet.add(`b:${card3}`);
            secondSet.add(`b:${card4}`);
            remainder.add(`c:${card5}`);
            [
              ...firstSet.toArray(),
              ...secondSet.toArray(),
              ...remainder.toArray(),
            ].forEach((c) => arrangedSet.add(c));
            if (arrangedSet.size() === 5) {
              if (arrangedSets.size() % 100 === 0) {
                console.log("Up to ", arrangedSets.size());
              }
              arrangedSets.add(arrangedSet.toArray().sort().join());
            }
          });
        });
      });
    });
  });
});

console.log("Sets", sets.size());
console.log("Arranged sets", arrangedSets.size());
console.log("Arranged sets sample", arrangedSets.toArray().slice(0, 100));