If `(a, 0), (0, b)` and `(1, 1)` are collinear, what is `(a+b-ab)` equal to ?

Options

A. 2
B. 1
C. 0
D. `-1`

Explanation

To solve the problem, we need to determine the value of \( a + b - ab \) given that the points \( (a, 0) \), \( (0, b) \), and \( (1, 1) \) are collinear.

### Step-by-Step Solution:

1. **Understanding Collinearity**:
Three points are collinear if the area of the triangle formed by them is zero. We can use the determinant method to find this area.

2. **Setting Up the Determinant**:
We can set up the determinant using the coordinates of the points:
\[ \begin{vmatrix} a & 0 & 1 \\ 0 & b & 1 \\ 1 & 1 & 1 \end{vmatrix} \]

3. **Calculating the Determinant**:
The determinant can be calculated as follows:
\[ = a \begin{vmatrix} b & 1 \\ 1 & 1 \end{vmatrix} - 0 + 1 \begin{vmatrix} 0 & b \\ 1 & 1 \end{vmatrix} \]
The first minor is:
\[ = a(b \cdot 1 - 1 \cdot 1) = a(b - 1) \]
The second minor is:
\[ = 0 \cdot 1 - b \cdot 1 = -b \]
Thus, the determinant simplifies to:
\[ a(b - 1) - b \]

4. **Setting the Determinant to Zero**:
For the points to be collinear, the determinant must equal zero:
\[ a(b - 1) - b = 0 \]

5. **Rearranging the Equation**:
Rearranging gives us:
\[ ab - a - b = 0 \]

6. **Finding the Relationship**:
From the equation \( ab - a - b = 0 \), we can rearrange it to find:
\[ ab = a + b \]

7. **Calculating \( a + b - ab \)**:
Now we substitute \( ab = a + b \) into \( a + b - ab \):
\[ a + b - ab = a + b - (a + b) = 0 \]

### Final Answer:
Thus, the value of \( a + b - ab \) is \( \boxed{0} \).