If `0le[x]lt2,-1le[y]lt1and1le[z]lt3`, where `[*]` denotes the greatest integer function, then the maximum value of the determinant
`|{:([x]+1,,[y],,[z]),([x],,[y]+1,,[z]),([x],,[y],,[z]+1):}|` is -

Options

A. 2
B. 4
C. 6
D. 8

Explanation

To solve the problem, we need to evaluate the determinant given the constraints on \(x\), \(y\), and \(z\). We will denote the greatest integer function as \([x]\), \([y]\), and \([z]\).

### Step-by-step Solution:

1. **Understanding the Constraints**:
- The constraints given are:
- \(0 \leq [x] < 2\)
- \(-1 \leq [y] < 1\)
- \(1 \leq [z] < 3\)

From these constraints, we can deduce the possible values for \([x]\), \([y]\), and \([z]\):
- \([x]\) can take values \(0\) or \(1\).
- \([y]\) can take values \(-1\) or \(0\).
- \([z]\) can take values \(1\) or \(2\).

2. **Setting Up the Determinant**:
The determinant we need to evaluate is:
\[ D = \begin{vmatrix} [x] + 1 & [y] & [z] \\ [x] & [y] + 1 & [z] \\ [x] & [y] & [z] + 1 \end{vmatrix} \]

3. **Calculating the Determinant**:
We will expand the determinant using the formula for a \(3 \times 3\) determinant:
\[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where:
- \(a = [x] + 1\), \(b = [y]\), \(c = [z]\)
- \(d = [x]\), \(e = [y] + 1\), \(f = [z]\)
- \(g = [x]\), \(h = [y]\), \(i = [z] + 1\)

Plugging in these values, we get:
\[ D = ([x] + 1)(([y] + 1)([z] + 1) - [z][y]) - [y]([x]([z + 1] - [z]) + [z]([x] - [x]) + [z]([y] - [y])) \]

4. **Evaluating for Maximum Values**:
We will evaluate \(D\) for all combinations of \([x]\), \([y]\), and \([z]\):
- For \([x] = 0\), \([y] = -1\), \([z] = 1\):
\[ D = (0 + 1)((-1 + 1)(1 + 1) - 1 \cdot (-1)) = 1(0 + 1) = 1 \]
- For \([x] = 1\), \([y] = 0\), \([z] = 2\):
\[ D = (1 + 1)((0 + 1)(2 + 1) - 2 \cdot 0) = 2(1 \cdot 3) = 6 \]
- For \([x] = 1\), \([y] = -1\), \([z] = 1\):
\[ D = (1 + 1)((-1 + 1)(1 + 1) - 1 \cdot (-1)) = 2(0 + 1) = 2 \]
- For \([x] = 0\), \([y] = 0\), \([z] = 2\):
\[ D = (0 + 1)((0 + 1)(2 + 1) - 2 \cdot 0) = 1(1 \cdot 3) = 3 \]

5. **Finding the Maximum Value**:
After evaluating all combinations, the maximum value of the determinant \(D\) is \(6\).

### Final Answer:
The maximum value of the determinant is \(6\).