The number of integers between `1` and `10000` with exactly one '4' and exactly one '5' as its digits does not lie in the interval :

Options

A. `[500, 600)`
B. `[600, 700)`
C. `[700, 800)`
D. `[800, 900)`

Explanation

To solve the problem of finding the number of integers between 1 and 10,000 that contain exactly one '4' and one '5', we can follow these steps:

### Step 1: Determine the total number of digits
The integers we are considering can have up to 4 digits (from 1 to 9999). Therefore, we will analyze 4-digit numbers, including leading zeros (e.g., 0001, 0002, ..., 9999).

### Step 2: Choose positions for '4' and '5'
We need to select 2 out of the 4 positions for the digits '4' and '5'. The number of ways to choose 2 positions from 4 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of positions and \( r \) is the number of positions to choose.

\[ \text{Number of ways to choose positions for '4' and '5'} = \binom{4}{2} = 6 \]

### Step 3: Arrange '4' and '5'
Once we have chosen the positions for '4' and '5', we can arrange them in those positions. Since we have one '4' and one '5', there are \( 2! \) (factorial of 2) ways to arrange these two digits.

\[ \text{Ways to arrange '4' and '5'} = 2! = 2 \]

### Step 4: Fill the remaining positions
After placing '4' and '5', we have 2 remaining positions to fill. The digits that can be used to fill these positions are from the set {0, 1, 2, 3, 6, 7, 8, 9} (all digits except '4' and '5'). This gives us 8 choices for each of the remaining positions.

\[ \text{Choices for remaining positions} = 8 \times 8 = 64 \]

### Step 5: Calculate the total combinations
Now, we can calculate the total number of integers that meet the criteria by multiplying the number of ways to choose positions, the arrangements of '4' and '5', and the choices for the remaining positions.

\[ \text{Total integers} = \binom{4}{2} \times 2! \times (8 \times 8) = 6 \times 2 \times 64 = 768 \]

### Conclusion
The total number of integers between 1 and 10,000 with exactly one '4' and one '5' is 768.