If `0le [x] lt 2, -1 le [y] lt 1 and 1le [z] lt3` where [.] denotes the greatest integer functions then the maximum value of `Delta` where `Delta=|([x]+1,[y],[z]),([x],[y]+1,[z]),([x],[y],[z]+1)|` is

Options

A. 2
B. 4
C. 6
D. none

Answer

To solve the problem, we need to find the maximum value of the determinant \( \Delta \) given the constraints on \( x \), \( y \), and \( z \) defined by their greatest integer functions.

### Step-by-Step Solution:

1. **Identify the ranges of the greatest integer functions**:
- For \( [x] \): \( 0 \leq [x] < 2 \) implies \( [x] \) can be \( 0 \) or \( 1 \).
- For \( [y] \): \( -1 \leq [y] < 1 \) implies \( [y] \) can be \( -1 \) or \( 0 \).
- For \( [z] \): \( 1 \leq [z] < 3 \) implies \( [z] \) can be \( 1 \) or \( 2 \).

2. **Set up the determinant**:
\[ \Delta = \left| \begin{array}{ccc} [x] + 1 & [y] & [z] \\ [x] & [y] + 1 & [z] \\ [x] & [y] & [z] + 1 \end{array} \right| \]

3. **Substituting possible values**:
- We will evaluate \( \Delta \) for all combinations of \( [x] \), \( [y] \), and \( [z] \).

4. **Calculate the determinant**:
Let's calculate \( \Delta \) for the combinations:
- **Case 1**: \( [x] = 0, [y] = -1, [z] = 1 \)
\[ \Delta = \left| \begin{array}{ccc} 1 & -1 & 1 \\ 0 & 0 & 1 \\ 0 & -1 & 2 \end{array} \right| \]
Expanding this determinant gives:
\[ \Delta = 1 \cdot \left| \begin{array}{cc} 0 & 1 \\ -1 & 2 \end{array} \right| = 1 \cdot (0 \cdot 2 - 1 \cdot -1) = 1 \]

- **Case 2**: \( [x] = 0, [y] = 0, [z] = 1 \)
\[ \Delta = \left| \begin{array}{ccc} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 2 \end{array} \right| \]
Expanding gives:
\[ \Delta = 1 \cdot \left| \begin{array}{cc} 1 & 1 \\ 0 & 2 \end{array} \right| = 1 \cdot (1 \cdot 2 - 1 \cdot 0) = 2 \]

- **Case 3**: \( [x] = 1, [y] = -1, [z] = 1 \)
\[ \Delta = \left| \begin{array}{ccc} 2 & -1 & 1 \\ 1 & 0 & 1 \\ 1 & -1 & 2 \end{array} \right| \]
Expanding gives:
\[ \Delta = 2 \cdot \left| \begin{array}{cc} 0 & 1 \\ -1 & 2 \end{array} \right| - (-1) \cdot \left| \begin{array}{cc} 1 & 1 \\ 1 & 2 \end{array} \right| \]
This results in:
\[ = 2(0 \cdot 2 - 1 \cdot -1) + (1 \cdot 2 - 1 \cdot 1) = 2 + 1 = 3 \]

- **Case 4**: \( [x] = 1, [y] = 0, [z] = 2 \)
\[ \Delta = \left| \begin{array}{ccc} 2 & 0 & 2 \\ 1 & 1 & 2 \\ 1 & 0 & 3 \end{array} \right| \]
Expanding gives:
\[ = 2 \cdot \left| \begin{array}{cc} 1 & 2 \\ 0 & 3 \end{array} \right| - 0 + 2 \cdot \left| \begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array} \right| \]
This results in:
\[ = 2(1 \cdot 3 - 2 \cdot 0) + 2(1 \cdot 0 - 1 \cdot 1) = 6 - 2 = 4 \]

5. **Find the maximum value**:
After evaluating all cases, the maximum value of \( \Delta \) is \( 4 \).

### Final Answer:
The maximum value of \( \Delta \) is \( \boxed{4} \).