<s>[INST] Distributing identical objects to identical boxes We have 6 identical things to be distributed in 4 identical boxes such that empty boxes are allowed the find the number of ways to distribute the things ?
 [/INST] The result is the number of partitions of 6 into a sum of 4 integral summands $\geq 0$.
The possibilities are
$$
6 = 6 + 0 + 0 + 0 \\
= 5 + 1 + 0 + 0 \\
= 4 + 2 + 0 + 0 \\
= 4 + 1 + 1 + 0 \\
= 3 + 3 + 0 + 0 \\
= 3 + 2 + 1 + 0 \\
= 3 + 1 + 1 + 1 \\
= 2 + 2 + 2 + 0 \\
= 2 + 2 + 1 + 1
$$
so the answer is $9$.
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