When writing all integers from 1 to 1000 inclusive, how many times in total does the digit 7 appear across all these numbers?

Introduction / Context:
Digit counting questions are popular in aptitude exams because they test both logical reasoning and systematic counting skills. Instead of manually writing and checking all numbers from 1 to 1000, we look for a clean method that uses symmetry and positional analysis. The goal here is to count how many times the digit 7 appears in all positions (units, tens, hundreds) when listing numbers from 1 up to 1000 inclusive.

Given Data / Assumptions:

• Numbers considered: 1, 2, 3, ..., 1000.
• We count every occurrence of the digit 7 in any position.
• The number 1000 does not contain the digit 7.
• Leading zeros are not written, but we may imagine numbers as three digit strings with leading zeros (from 001 to 999) for easier counting.
• All standard decimal representations are used without skipping any number.

Concept / Approach:
A common approach is to treat 001 to 999 as three digit numbers and analyze each position (hundreds, tens, units) separately. Due to symmetry, the digit 7 appears equally often in each position. Once we know how many times 7 appears in one position, we can multiply by 3 to get the total from 001 to 999. Finally, we check whether 1000 adds any extra 7, which it does not.

Step-by-Step Solution:
Step 1: Consider numbers from 000 to 999 as all three digit strings. Step 2: There are 1000 such strings (from 000 to 999). Step 3: For the units place, each digit from 0 to 9 appears equally often. Step 4: Therefore, digit 7 appears 1000 / 10 = 100 times in the units place. Step 5: The same logic applies to the tens place: 7 appears 100 times in the tens position. Step 6: Similarly, in the hundreds place, 7 appears 100 times. Step 7: Total occurrences from 000 to 999 = 100 + 100 + 100 = 300. Step 8: Now ignore 000 (not part of 1 to 1000). This does not affect the count of digit 7. Step 9: Check 1000, which has no 7, so the total remains 300.

Verification / Alternative check:
You can make a quick partial check by counting 7s from 1 to 99 and seeing the pattern. From 0 to 99, the digit 7 appears 20 times in the units place and 20 times in the tens place for a total of 40. Extending this logic up to 999 results in 300 total occurrences, which matches our positional reasoning. Computer based checks also confirm this total.

Why Other Options Are Wrong:
243: This corresponds to 3 * 9^2, which is not related to the correct equal distribution logic. 301: Just one more than the correct value, probably from miscounting 1000 as containing a 7 somewhere. 290: A rounded guess that ignores the exact distribution across all positions. Only 300 fits the precise positional counting argument.

Common Pitfalls:
A frequent error is to try to count manually in ranges and accidentally double count or skip numbers. Another mistake is forgetting about the hundreds place or misinterpreting leading zeros. Some students mistakenly think that each digit appears only 90 times instead of 100 in one position due to confusion between ranges. Using the structured approach of considering 000 to 999 and then adjusting for 1 to 1000 makes the problem much easier and less error prone.

Final Answer:
The digit 7 is written a total of 300 times when listing all integers from 1 to 1000 inclusive.