Let n be 4-digit integer in which all the digits are different. If x is the number of odd integers and y is the number of even integers, then

Options

A. `x lt y`
B. `x gt y`
C. `x+y=4500`
D. `|x-y|=56`

Explanation

To solve the problem, we need to find the number of 4-digit integers where all digits are different, and then determine the counts of odd integers (x) and even integers (y) based on the last digit.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We need to find 4-digit integers where all digits are different.
- The integers can either be odd or even, depending on the last digit.

2. **Counting Odd Integers (x)**:
- A 4-digit integer is odd if its last digit is one of the odd digits: 1, 3, 5, 7, or 9.
- There are 5 choices for the last digit (unit place).
- After choosing the last digit, we have 3 more positions (thousands, hundreds, and tens) to fill with different digits.

3. **Choosing the First Digit**:
- The first digit (thousands place) cannot be 0 and cannot be the digit already used in the unit place.
- Therefore, we have 8 options left for the first digit (from the digits 1-9 excluding the chosen odd digit).

4. **Choosing the Remaining Digits**:
- After selecting the first digit, we have 8 digits remaining (including 0) for the hundreds place.
- After selecting the hundreds place digit, we have 7 choices left for the tens place.

5. **Calculating the Total for Odd Integers**:
- The total number of odd integers (x) can be calculated as:
\[ x = \text{(choices for unit place)} \times \text{(choices for thousands place)} \times \text{(choices for hundreds place)} \times \text{(choices for tens place)} \]
\[ x = 5 \times 8 \times 8 \times 7 = 2240 \]

6. **Counting Even Integers (y)**:
- A 4-digit integer is even if its last digit is one of the even digits: 0, 2, 4, 6, or 8.
- If the last digit is 0, we have 9 choices for the first digit (1-9) and then 8 for the hundreds place and 7 for the tens place:
\[ \text{If last digit is 0: } 9 \times 8 \times 7 = 504 \]
- If the last digit is one of the other even digits (2, 4, 6, 8), we have 4 choices for the last digit.
- For the first digit, we have 8 options (1-9 excluding the chosen even digit).
- For the hundreds place, we have 8 options (including 0).
- For the tens place, we have 7 options.
\[ \text{If last digit is 2, 4, 6, or 8: } 4 \times 8 \times 8 \times 7 = 1792 \]

7. **Calculating the Total for Even Integers**:
- The total number of even integers (y) is:
\[ y = 504 + 1792 = 2296 \]

8. **Comparing x and y**:
- Now we compare x and y:
- \( x = 2240 \)
- \( y = 2296 \)
- Therefore, \( x < y \).

9. **Final Results**:
- We can summarize our findings:
- \( x < y \) is true.
- \( x + y \neq 4500 \).
- \( |x - y| = 56 \) (which is not equal to 0).

### Conclusion:
- The correct statements are:
- \( x < y \)