1. **Stating the problem:** We have 16 volleyball teams divided into 3 ranking tiers: 4 teams in 1st tier, 4 teams in 2nd tier, and 8 teams in 3rd tier. 2. The 4 teams of the 1st tier are assigned directly to groups A, B, C, and D respectively, so no choice here. 3. For each group, we must add 1 team from the 2nd tier and 2 teams from the 3rd tier. 4. We want to find the total number of possible compositions of the 4 groups under these constraints. 5. **Formula and reasoning:** - The 4 teams of 2nd tier must be distributed one per group. The number of ways to assign 4 distinct 2nd tier teams to 4 distinct groups is the number of permutations of 4 elements: $$4! = 24$$. - The 8 teams of 3rd tier must be divided into 4 groups of 2 teams each. - The number of ways to split 8 distinct teams into 4 groups of 2 (order of groups matters, order within groups does not) is given by: $$\frac{8!}{(2!)^4}$$ Explanation: We arrange all 8 teams in order ($8!$), then divide by $(2!)^4$ because within each pair the order does not matter. 6. **Total number of compositions:** $$\text{Total} = 4! \times \frac{8!}{(2!)^4}$$ 7. **Calculate intermediate values:** - $4! = 24$ - $8! = 40320$ - $(2!)^4 = (2)^4 = 16$ 8. Substitute values: $$\text{Total} = 24 \times \frac{40320}{16}$$ 9. Simplify fraction: $$\frac{40320}{16} = 2520$$ 10. Multiply: $$24 \times 2520 = 60480$$ **Final answer:** $$\boxed{60480}$$ There are 60480 possible compositions of the groups A, B, C, D under the given constraints.