Consider the nine digit number n = 7 3 `alpha` 4 9 6 1 `beta` 0. If p is th number of all possible distinct values of `(alpha-beta)`, then P is equal to

Options

A. 17
B. 18
C. 19
D. 20

Answer

To solve the problem, we need to determine the distinct values of the expression \( \alpha - \beta \) where \( \alpha \) and \( \beta \) are digits ranging from 0 to 9. The nine-digit number is given as \( n = 73\alpha4961\beta0 \).

### Step-by-Step Solution:

1. **Identify the Range of Values for \( \alpha \) and \( \beta \)**:
- Since \( \alpha \) and \( \beta \) are digits, they can take values from 0 to 9.

2. **Determine the Expression \( \alpha - \beta \)**:
- The expression \( \alpha - \beta \) can yield different results based on the values of \( \alpha \) and \( \beta \).

3. **Calculate the Maximum and Minimum Values of \( \alpha - \beta \)**:
- The maximum value occurs when \( \alpha \) is at its highest (9) and \( \beta \) is at its lowest (0):
\[ \text{Maximum} = 9 - 0 = 9 \]
- The minimum value occurs when \( \alpha \) is at its lowest (0) and \( \beta \) is at its highest (9):
\[ \text{Minimum} = 0 - 9 = -9 \]

4. **List All Possible Values of \( \alpha - \beta \)**:
- The values of \( \alpha - \beta \) can range from -9 to 9. This includes:
\[ -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \]

5. **Count the Distinct Values**:
- The distinct values from -9 to 9 include all integers in this range. To count them:
- From -9 to 9, there are a total of:
\[ 9 - (-9) + 1 = 9 + 9 + 1 = 19 \]

6. **Conclusion**:
- The number of all possible distinct values of \( \alpha - \beta \) is \( p = 19 \).

### Final Answer:
Thus, \( P \) is equal to \( 19 \).