Nine gifts, three of which are identical and the rest are distinct, are distributed among five people without restrictions on the number of gifts a person can have. By first considering the number of ways to distribute the distinct gifts or otherwise, find the number of way that the nine gifts can be distributed.

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 = $15625 \times 35 = 546875$.