To solve the problem of distributing four different balls (alpha, beta, gamma, and delta) into four different boxes (A, B, C, and D) with the given restrictions, we can follow these steps:
### Step 1: Identify the restrictions
- Ball alpha cannot go into box A.
- Ball beta cannot go into box B.
### Step 2: Determine the options for each box
- **Box A**: Since ball alpha cannot go into box A, we have three options for box A: beta, gamma, or delta.
- **Box B**: Since ball beta cannot go into box B, we also have three options for box B: alpha, gamma, or delta.
- **Box C**: There are no restrictions for box C, so we can place any of the four balls (alpha, beta, gamma, delta) in box C.
- **Box D**: Similarly, there are no restrictions for box D, so we can place any of the four balls (alpha, beta, gamma, delta) in box D.
### Step 3: Calculate the number of ways to fill the boxes
- For box A, we have 3 choices (beta, gamma, delta).
- For box B, we also have 3 choices (alpha, gamma, delta).
- For box C, we have 4 choices (alpha, beta, gamma, delta).
- For box D, we have 4 choices (alpha, beta, gamma, delta).
### Step 4: Multiply the number of choices
The total number of ways to fill the boxes while adhering to the restrictions is given by multiplying the number of choices for each box:
\[
\text{Total Ways} = (\text{Choices for A}) \times (\text{Choices for B}) \times (\text{Choices for C}) \times (\text{Choices for D})
\]
Substituting the values we found:
\[
\text{Total Ways} = 3 \times 3 \times 4 \times 4
\]
### Step 5: Calculate the final result
Calculating the above expression:
\[
\text{Total Ways} = 3 \times 3 = 9
\]
\[
9 \times 4 = 36
\]
\[
36 \times 4 = 144
\]
Thus, the total number of ways to fill the boxes is **144**.
### Final Answer
The number of ways of filling the balls in the boxes such that each box contains one ball, with the given restrictions, is **144**.
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