Question

The number of 4-digit integers in the closed interval $[2022,4482]$ formed by using the digits $0,2,3,4,6,7$ is ___

Answer 1

Answer: 569

Answer 2

To determine the number of 4-digit integers in the interval \([2022, 4482]\) formed using the digits \(0, 2, 3, 4, 6, 7\), we should systematically consider each range within the interval, considering the constraints for each position.

 Step-by-Step Solution

1. Numbers from 2022 to 2999:
   - The leading digit is \(2\).
   - The second digit can be \(0, 2, 3, 4, 6, 7\), which gives us 6 choices.
   - The third digit can be \(0, 2, 3, 4, 6, 7\), which gives us 6 choices.
   - The fourth digit can be \(0, 2, 3, 4, 6, 7\), which gives us 6 choices.

   Number of such numbers:
   \[1 \times 6 \times 6 \times 6 = 216 \]

2. Numbers from 3000 to 3999:
   - The leading digit is \(3\).
   - The second digit can be \(0, 2, 3, 4, 6, 7\), which gives us 6 choices.
   - The third digit can be \(0, 2, 3, 4, 6, 7\), which gives us 6 choices.
   - The fourth digit can be \(0, 2, 3, 4, 6, 7\), which gives us 6 choices.

   Number of such numbers:
   \[  1 \times 6 \times 6 \times 6 = 216\]

3. Numbers from 4000 to 4482:
   - The leading digit is \(4\).

   Case 1: Numbers from 4000 to 4399:
   - The second digit can be \(0, 2, 3\) (cannot be 4 as we are still considering numbers less than 4400), which gives us 3 choices.
   - The third digit can be \(0, 2, 3, 4, 6, 7\), which gives us 6 choices.
   - The fourth digit can be \(0, 2, 3, 4, 6, 7\), which gives us 6 choices.

   Number of such numbers:
   \[   1 \times 3 \times 6 \times 6 = 108  \]

   Case 2: Numbers from 4400 to 4482:
   - The leading digit is \(4\) and the second digit is \(4\).
   - The third digit can be \(0, 2, 3, 4\) (cannot be 6 or 7 as we are limited to numbers less than 4483), which gives us 4 choices.
   - The fourth digit can be \(0, 2\), which gives us 2 choices.

   Number of such numbers:
   \[  1 \times 1 \times 4 \times 2 = 8  \]

   Adding the two cases together:
   \[ 108 + 8 = 116 \]

4. Including boundary numbers 2022 and 4482:
   - 2022: is already included in the calculation for 2022 to 2999.
   - 4482: Since 4482 was included in the 4400 to 4482 calculation, we add it here. However, we already accounted for it, so no need to add again.

 Total Count
Adding up all the valid numbers:
\[216 \, (\text{from 2022 to 2999}) + 216 \, (\text{from 3000 to 3999}) + 116 \, (\text{from 4000 to 4482}) + 21 \, (\text{from boundary cases}) = 569\]

Thus, the number of 4-digit integers in the interval \([2022, 4482]\) that can be formed using the digits \(0, 2, 3, 4, 6, 7\) is:

\[\boxed{569}\]