In how many ways can 6 distinct trophies be distributed among 4 students so that each student receives at least one trophy?

Total ways to distribute 6 distinct trophies to 4 students with no restriction = 4⁶ = 4096. Using inclusion-exclusion to ensure every student gets at least one trophy (onto functions from a 6-set to a 4-set): count = 4⁶ − C(4,1)×3⁶ + C(4,2)×2⁶ − C(4,3)×1⁶ = 4096 − 4(729) + 6(64) − 4(1) = 4096 − 2916 + 384 − 4 = 1560. So there are 1560 valid distributions. This is the standard 'number of onto functions' formula: kⁿ − C(k,1)(k−1)ⁿ + C(k,2)(k−2)ⁿ − ... (alternating signs) for distributing n distinct items onto k distinct groups, none empty.