A girl wrote all the numbers from 100 to 200. Then she started counting the number of one's that has been used while writing all these numbers. What is the number that she got?

Step 1 (Formulate digit position analysis): The girl wrote all integers from 100 to 200. To count the total number of 1's used, we analyze the occurrence of 1 in each digit position:

• Case 1: Hundreds Place ($1XX$):

The digit 1 appears in the hundreds place in all numbers from $100$ to $199$. There are exactly 100 numbers in this range. For the number 200, the hundreds digit is 2. Total = 100.

• Case 2: Tens Place ($X1X$):

The digit 1 appears in the tens place in numbers of the form $110$ to $119$. There are exactly 10 such numbers. Total = 10.

• Case 3: Units Place ($XX1$):

The digit 1 appears in the units place in numbers of the form $101,111,121,131,141,151,161,171,181,191$. There are exactly 10 such numbers. Total = 10.

Step 2 (Sum the digit counts): We add the three counts together:

Step 3 (Conclusion): The number of ones used is exactly 120.