Question

How many numbers \( x \) exist such that \( 10^{11}<x<10^{12} \) and the sum of digits of \( x \) is 1?

Hint:

For sum of digits problems, especially when the sum is small, consider numbers where one digit is non-zero and all others are zero.

Answer 1

Step 1: Understanding the range.
We are asked to find how many numbers \( x \) exist such that \( 10^{11}<x<10^{12} \). This means that \( x \) is a 12-digit number. Step 2: Condition on sum of digits.
The sum of digits of \( x \) is 1. The only way a 12-digit number can have a sum of digits equal to 1 is if one of its digits is 1 and all other digits are 0. Step 3: Identifying the possibilities.
Since \( x \) is a 12-digit number, the digit 1 can appear in any one of the 12 positions, with the remaining 11 digits being 0. Therefore, there are 12 possible positions for the digit 1. Step 4: Conclusion.
Thus, there are 9 numbers that satisfy the condition \( 10^{11}<x<10^{12} \) with a sum of digits equal to 1, and the correct answer is 9.