If `alpha=x_1,x_2,x_3 and beta=y_1,y_2,y_3` be two three digit numbers, then the number of pairs of `alpha and beta` that can be formed so that `alpha` can be subtracted from `beta` without borrowing.

Options

A. `2!xx10!xx10!`
B. `(45)(55)^(2)`
C. `3^(2)*5^(3)*11^(2)`
D. `136125

Explanation

To solve the problem of finding the number of pairs of three-digit numbers \( \alpha = x_1 x_2 x_3 \) and \( \beta = y_1 y_2 y_3 \) such that \( \beta \) can be subtracted from \( \alpha \) without borrowing, we will follow these steps:

### Step 1: Understand the Conditions
1. **Three-Digit Numbers**: Both \( \alpha \) and \( \beta \) are three-digit numbers, meaning \( x_1, y_1 \) cannot be 0. Thus, \( x_1, y_1 \) can take values from 1 to 9.
2. **No Borrowing Condition**: For subtraction without borrowing, each digit of \( \beta \) must be greater than or equal to the corresponding digit of \( \alpha \):
- \( y_1 \geq x_1 \)
- \( y_2 \geq x_2 \)
- \( y_3 \geq x_3 \)

### Step 2: Calculate Possible Values for Each Digit
1. **For \( x_1 \) and \( y_1 \)**:
- Possible values for \( x_1 \): 1 to 9 (9 options)
- For each value of \( x_1 = \lambda \) (where \( \lambda \) can be from 1 to 9), \( y_1 \) can take values from \( \lambda \) to 9.
- The number of options for \( y_1 \) is \( 10 - \lambda \).
- Total pairs for \( x_1, y_1 \) is:
\[ \sum_{\lambda=1}^{9} (10 - \lambda) = 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 45 \]

2. **For \( x_2 \) and \( y_2 \)**:
- Possible values for \( x_2 \): 0 to 9 (10 options)
- For each value of \( x_2 = \lambda \) (where \( \lambda \) can be from 0 to 9), \( y_2 \) can take values from \( \lambda \) to 9.
- The number of options for \( y_2 \) is \( 10 - \lambda \).
- Total pairs for \( x_2, y_2 \) is:
\[ \sum_{\lambda=0}^{9} (10 - \lambda) = 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 55 \]

3. **For \( x_3 \) and \( y_3 \)**:
- The calculation is identical to that of \( x_2 \) and \( y_2 \).
- Total pairs for \( x_3, y_3 \) is also:
\[ \sum_{\lambda=0}^{9} (10 - \lambda) = 55 \]

### Step 3: Calculate Total Pairs
The total number of pairs \( (\alpha, \beta) \) is the product of the pairs calculated for each digit:
\[ \text{Total pairs} = 45 \times 55 \times 55 = 45 \times 55^2 \]

### Final Answer
Thus, the number of pairs \( (\alpha, \beta) \) that can be formed such that \( \alpha \) can be subtracted from \( \beta \) without borrowing is:
\[ \boxed{45 \times 55^2} \]