将九件礼物分给五个人，其中三件完全相同，其余各不相同，且对每个人可获得的礼物数量没有限制。通过首先考虑分配不同礼物的方法数或其他方法，求分配这九件礼物的方案数。

1. Calculate the number of ways to distribute the 6 distinct gifts.

The number of distinct gifts is $9 - 3 = 6$.
Since there are no restrictions, each of the $6$ distinct gifts can be given to any of the $5$ people.
Number of ways to distribute the distinct gifts = $5^6 = 15625$.

2. Calculate the number of ways to distribute the 3 identical gifts.

Since the $3$ gifts are identical, we only care about how many gifts each person receives. We can consider the possible groupings:

• All 3 gifts to 1 person: Choose $1$ person out of $5$ to receive all of them. This gives $\binom{5}{1} = 5$ ways.
• 2 gifts to 1 person, 1 gift to another: Choose $1$ person to receive $2$ gifts ($5$ ways), and $1$ of the remaining $4$ people to receive $1$ gift ($4$ ways). This gives $5 \times 4 = 20$ ways.
• 1 gift each to 3 different people: Choose $3$ people out of $5$ to receive $1$ gift each. This gives $\binom{5}{3} = 10$ ways.

Total number of ways to distribute the identical gifts = $5 + 20 + 10 = 35$.

(Alternatively, using the 'stars and bars' combinations with repetition formula: $\binom{3+5-1}{5-1} = \binom{7}{4} = 35$.)

3. Calculate the overall total number of ways to distribute all the gifts.

By the multiplication principle, the distinct and identical gifts can be distributed independently.
Total number of ways = $15625 \times 35 = 546875$.