To solve the problem, we need to determine the value of \( x \) such that the mode of the data set \( 4, 3, 2, 5, x, 4, 5, 1, 7, 3, 2, 1 \) is 4. The mode is the number that appears most frequently in a data set.
### Step-by-Step Solution:
1. **Understand the Mode**: The mode is the value that appears most frequently in the dataset. According to the problem, the mode is given as 4.
2. **Count the Frequency of Existing Numbers**:
- The numbers in the dataset are: \( 4, 3, 2, 5, x, 4, 5, 1, 7, 3, 2, 1 \).
- Let's count the occurrences of each number without considering \( x \):
- \( 4 \): appears 2 times
- \( 3 \): appears 2 times
- \( 2 \): appears 2 times
- \( 5 \): appears 2 times
- \( 1 \): appears 2 times
- \( 7 \): appears 1 time
3. **Determine the Impact of \( x \)**: Since the mode is 4, for 4 to be the mode, it must appear more frequently than any other number. Currently, 4 appears 2 times. To make 4 the mode, it needs to appear at least 3 times.
4. **Set Up the Equation**:
- If \( x \) is equal to 4, then the frequency of 4 will increase to 3.
- Thus, the frequency of 4 will be: \( 2 + 1 = 3 \) (if \( x = 4 \)).
5. **Check the Frequencies**:
- If \( x = 4 \):
- \( 4 \): 3 times
- \( 3 \): 2 times
- \( 2 \): 2 times
- \( 5 \): 2 times
- \( 1 \): 2 times
- \( 7 \): 1 time
- The number 4 appears the most (3 times), confirming that it is the mode.
6. **Conclusion**: Therefore, the value of \( x \) must be 4 for the mode of the data set to be 4.
### Final Answer:
The value of \( x \) is **4**.
