1. **Problem statement:** We want to find the number of ways to distribute six indistinguishable objects into four indistinguishable boxes such that each box contains at least one object. 2. **Key concept:** Since both objects and boxes are indistinguishable, this is a problem of counting partitions of the integer 6 into exactly 4 positive parts. 3. **Mathematical formulation:** We need to find the number of integer solutions to $$x_1 + x_2 + x_3 + x_4 = 6$$ where $x_i \geq 1$ and the order of $x_i$ does not matter (because boxes are indistinguishable). 4. **Counting partitions:** The problem reduces to counting the partitions of 6 into 4 positive parts. The partitions of 6 into 4 parts are: - 3 + 1 + 1 + 1 - 2 + 2 + 1 + 1 - 2 + 1 + 1 + 1 + 1 (not valid since only 4 parts allowed) - 1 + 1 + 1 + 3 (same as first, order does not matter) Actually, the distinct partitions of 6 into 4 positive parts are: - 3 + 1 + 1 + 1 - 2 + 2 + 1 + 1 - 2 + 1 + 1 + 2 (same as above) So the unique partitions are: - 3 + 1 + 1 + 1 - 2 + 2 + 1 + 1 5. **List all partitions explicitly:** - (3,1,1,1) - (2,2,1,1) 6. **Count:** There are exactly 2 such partitions. **Final answer:** $$\boxed{2}$$