The exact number of integers greater than 800 and less than 4000 that can be formed using the digits 0, 1, 2, 4, 5, 8, 9 is 783.
Understanding the Problem and Available Digits
This problem requires us to count numbers within a specific range (801 to 3999, inclusive) using only a given set of digits. The allowed digits are {0, 1, 2, 4, 5, 8, 9}, which comprise a total of 7 unique digits.
We need to consider numbers that are either three-digit or four-digit numbers, as per the specified range:
- Three-digit numbers: from 801 to 999.
- Four-digit numbers: from 1000 to 3999.
Calculating Three-Digit Numbers
For numbers between 801 and 999, the first digit (hundreds place) must be either 8 or 9. The remaining two digits (tens and units place) can be any of the 7 allowed digits.
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Numbers starting with 8 (8XX):
- The hundreds digit is fixed as 8 (1 choice).
- The tens digit can be any of the 7 available digits (0, 1, 2, 4, 5, 8, 9).
- The units digit can be any of the 7 available digits.
- This forms 1 × 7 × 7 = 49 potential numbers (e.g., 800, 801, ..., 899).
- Since the numbers must be greater than 800, we must exclude 800 itself.
- Therefore, valid numbers starting with 8 are 49 - 1 = 48.
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Numbers starting with 9 (9XX):
- The hundreds digit is fixed as 9 (1 choice).
- The tens digit can be any of the 7 available digits.
- The units digit can be any of the 7 available digits.
- This forms 1 × 7 × 7 = 49 potential numbers (e.g., 900, 901, ..., 999). All these numbers are greater than 800.
Total three-digit numbers: 48 + 49 = 97.
Calculating Four-Digit Numbers
For numbers between 1000 and 3999, the first digit (thousands place) must be 1, 2, or 3. The remaining three digits can be any of the 7 allowed digits.
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Numbers starting with 1 (1XXX):
- The thousands digit is fixed as 1 (1 choice).
- The hundreds, tens, and units digits can each be any of the 7 available digits.
- This forms 1 × 7 × 7 × 7 = 343 numbers.
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Numbers starting with 2 (2XXX):
- The thousands digit is fixed as 2 (1 choice).
- The hundreds, tens, and units digits can each be any of the 7 available digits.
- This forms 1 × 7 × 7 × 7 = 343 numbers.
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Numbers starting with 3 (3XXX):
- The thousands digit would need to be 3.
- However, the digit '3' is not included in our allowed set of digits {0, 1, 2, 4, 5, 8, 9}.
- Therefore, 0 numbers can be formed starting with 3.
Total four-digit numbers: 343 + 343 + 0 = 686.
Summary of Counts
The total number of valid integers is the sum of the valid three-digit and four-digit numbers:
| Category | Details | Count |
|---|---|---|
| Three-Digit Numbers | ||
| Numbers from 801-899 | (1 choice for '8') × (7 choices) × (7 choices) - 1 (for 800) | 48 |
| Numbers from 900-999 | (1 choice for '9') × (7 choices) × (7 choices) | 49 |
| Subtotal (3-Digit) | 97 | |
| Four-Digit Numbers | ||
| Numbers from 1000-1999 | (1 choice for '1') × (7 choices)^3 | 343 |
| Numbers from 2000-2999 | (1 choice for '2') × (7 choices)^3 | 343 |
| Numbers from 3000-3999 | (Digit '3' is not available) | 0 |
| Subtotal (4-Digit) | 686 | |
| Grand Total | Sum of 3-Digit and 4-Digit Numbers | 783 |
By combining the counts for three-digit and four-digit numbers, we arrive at the final answer.
