In a digit counting problem, you write all the integers from 121 to 1346 inclusive on a typewriter. If each digit typed counts as one key press, how many digit key presses are required in total to type all these numbers?

Introduction / Context:
This question tests your understanding of place value and counting techniques. It asks for the total number of digit key presses needed to type all numbers in a given range, which is a standard type of aptitude puzzle. Rather than writing out each number individually, you should break the interval into groups with the same number of digits and then multiply by the number of digits per number.

Given Data / Assumptions:
- We are typing all integers from 121 up to 1346, inclusive at both ends.
- Each digit typed is one key press, and we ignore spaces, commas, or any other formatting characters.
- Numbers between 121 and 999 are three digit numbers, and numbers from 1000 to 1346 are four digit numbers.

Concept / Approach:
The method is to split the range into two continuous sub ranges where the number of digits is constant. From 121 to 999, every number has three digits. From 1000 to 1346, every number has four digits. We count how many numbers lie in each sub range, multiply by the number of digits in each, and then add the results. This avoids dealing with each number individually and makes the calculation efficient and accurate.

Step-by-Step Solution:
1) First consider the three digit numbers from 121 to 999 inclusive. 2) The count of these numbers is 999 − 121 + 1 = 879. 3) Each of these numbers has 3 digits, so the total digit presses for this part is 879 · 3 = 2637. 4) Next consider the four digit numbers from 1000 to 1346 inclusive. 5) The count of these numbers is 1346 − 1000 + 1 = 347. 6) Each has 4 digits, so the total digit presses for this part is 347 · 4 = 1388. 7) Add the two parts: 2637 + 1388 = 4025 digit key presses in total.

Verification / Alternative check:
We can quickly check the boundaries. From 121 to 999, we start at the first three digit number greater than or equal to 121 and go up to the last three digit number. The formula 999 − 121 + 1 is standard for inclusive ranges. Similarly, for four digit numbers, 1000 to 1346 inclusive gives 347 numbers. Multiplying and adding again gives 2637 + 1388, and the sum 4025 is consistent with both arithmetic and the structure of the number line.

Why Other Options Are Wrong:
Options A (4018), B (4021), and E (4035) are very close and arise from small counting mistakes, such as off by one errors at boundaries or mis multiplying 879 or 347. Option C (3675) indicates a more serious error, possibly counting only part of the range or using the wrong number of digits per group. Only Option D exactly matches the total obtained by careful grouping and arithmetic.

Common Pitfalls:
A common error is to forget that both endpoints are included, leading to subtracting without adding one. Another mistake is to forget that 1000 introduces the four digit range, causing all numbers to be treated as three digit numbers. Students also sometimes mis calculate products like 879 · 3, especially under time pressure. Keeping track of the digit length per range and checking simple multiplication carefully helps avoid these pitfalls.

Final Answer:
The total number of digit key presses required to type all integers from 121 to 1346 inclusive is 4025.