To solve the problem of distributing 4 identical red balls, 1 white ball, 1 green ball, and 1 black ball among 4 persons such that each person receives at least one ball and no one gets all identical red balls, we can follow these steps:
### Step 1: Distributing Red Balls
We have 4 identical red balls to distribute among 4 persons (let's denote them as A, B, C, and D). Since no one can receive all 4 red balls, we can distribute them in the following valid ways:
1. **Distribution of (1, 1, 1, 1)**: Each person gets 1 red ball.
2. **Distribution of (1, 1, 2, 0)**: Two persons get 1 red ball each, and one person gets 2 red balls (the fourth person gets none).
### Step 2: Calculating Ways for Distribution (1, 1, 1, 1)
In this case, there is only 1 way to distribute the red balls since each person gets exactly 1 ball.
### Step 3: Calculating Ways for Distribution (1, 1, 2, 0)
Here, we need to choose 2 persons to receive 1 red ball each and 1 person to receive 2 red balls. The fourth person will receive none.
- The number of ways to choose 3 persons from 4 is given by \( \binom{4}{3} = 4 \).
- For each selection of 3 persons, there are 3 ways to assign who gets 2 red balls.
Thus, the total ways for this distribution is:
\[
4 \times 3 = 12
\]
### Step 4: Distributing Other Balls
Now, we need to distribute the white, green, and black balls. Each of these balls can go to any of the 4 persons, and since there is only 1 of each color, each can be distributed in 4 ways.
- For the white ball: 4 choices
- For the green ball: 4 choices
- For the black ball: 4 choices
The total ways to distribute these balls is:
\[
4 \times 4 \times 4 = 64
\]
### Step 5: Total Ways for Distribution (1, 1, 1, 1) and (1, 1, 2, 0)
Now we combine the distributions:
1. For (1, 1, 1, 1):
\[
1 \times 64 = 64
\]
2. For (1, 1, 2, 0):
\[
12 \times 64 = 768
\]
Thus, the total number of ways \( n \) is:
\[
n = 64 + 768 = 832
\]
### Step 6: Finding \( \frac{n}{104} \)
Now, we need to find \( \frac{n}{104} \):
\[
\frac{832}{104} = 8
\]
### Final Answer
The value of \( \frac{n}{104} \) is **8**.
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