To solve the problem step by step, we need to analyze the given numbers X, Y, Z, and T, where the digit 7 is used only once in each number. The goal is to determine which of these four numbers is the greatest.
### Step 1: Understand the structure of the numbers
The numbers are structured as follows:
- X: `** 7 ** **`
- Y: `7 ** ** **`
- Z: `** ** 7 **`
- T: `** ** ** 7`
Here, `*` represents any digit from 0 to 6 (since 7 is used only once).
### Step 2: Identify the position of the digit 7
The position of the digit 7 in each number is crucial because the value of a number is determined by its leftmost digits. The more significant the position of a digit, the greater its impact on the overall value of the number.
- In Y, 7 is in the thousands place, making it the largest possible value.
- In X, Z, and T, 7 is in less significant positions (hundreds, tens, and units).
### Step 3: Compare the possible values
1. **Y**: Since 7 is in the thousands place, the smallest value Y can take is 7000 (if the other digits are all 0).
2. **X**: The highest value X can take is 6799 (if the other digits are 6, 9, and 9).
3. **Z**: The highest value Z can take is 9679 (if the other digits are 6, 9, and 9).
4. **T**: The highest value T can take is 9967 (if the other digits are 9, 9, and 6).
### Step 4: Determine the maximum possible values
- Y: Minimum value = 7000
- X: Maximum value = 6799
- Z: Maximum value = 9679
- T: Maximum value = 9967
### Step 5: Conclusion
From the analysis, we see that Y is the only number that can reach at least 7000, while all other numbers (X, Z, T) cannot exceed 9967. However, since Y starts at 7000, it is the greatest number among the four.
Thus, the greatest number among X, Y, Z, and T is **Y**.
