How many times does the digit 3 appear while writing the integers from 1 to 1000

Options

A. 269
B. 308
C. 300
D. None of these

Answer

To find out how many times the digit 3 appears while writing the integers from 1 to 1000, we can break down the problem into counting the occurrences of the digit 3 in each digit place (hundreds, tens, and units) for the numbers from 000 to 999.

### Step-by-Step Solution:

1. **Understanding the Range**:
We are looking at the numbers from 1 to 1000. However, for ease of calculation, we can consider the numbers from 000 to 999, and then we will account for the number 1000 separately.

2. **Counting Occurrences in Each Digit Place**:
Each number can be represented as a three-digit number (xyz), where x is the hundreds place, y is the tens place, and z is the units place.

3. **Counting 3 in the Hundreds Place**:
- The hundreds place can be 3 for the numbers from 300 to 399.
- There are 100 numbers in this range (300, 301, ..., 399).
- Therefore, the digit 3 appears **100 times** in the hundreds place.

4. **Counting 3 in the Tens Place**:
- The tens place can be 3 for the numbers: 030 to 039, 130 to 139, 230 to 239, 330 to 339, 430 to 439, 530 to 539, 630 to 639, 730 to 739, 830 to 839, and 930 to 939.
- Each of these ranges contains 10 numbers, and there are 10 such ranges (0-9 in the hundreds place).
- Therefore, the digit 3 appears **10 x 10 = 100 times** in the tens place.

5. **Counting 3 in the Units Place**:
- The units place can be 3 for the numbers: 003, 013, 023, 033, 043, 053, 063, 073, 083, 093, 103, 113, 123, 133, 143, 153, 163, 173, 183, 193, 203, 213, 223, 233, 243, 253, 263, 273, 283, 293, 303, 313, 323, 333, 343, 353, 363, 373, 383, 393, 403, 413, 423, 433, 443, 453, 463, 473, 483, 493, 503, 513, 523, 533, 543, 553, 563, 573, 583, 593, 603, 613, 623, 633, 643, 653, 663, 673, 683, 693, 703, 713, 723, 733, 743, 753, 763, 773, 783, 793, 803, 813, 823, 833, 843, 853, 863, 873, 883, 893, 903, 913, 923, 933, 943, 953, 963, 973, 983, 993.
- There are 10 occurrences for each hundred (0-9), and since there are 10 hundreds (0-9), the digit 3 appears **10 x 10 = 100 times** in the units place.

6. **Total Count**:
- Now, we sum the occurrences from all three places:
- Hundreds place: 100
- Tens place: 100
- Units place: 100
- Total = 100 + 100 + 100 = **300 times**.

7. **Considering the Number 1000**:
- The number 1000 does not contain the digit 3, so it does not affect our count.

### Final Answer:
The digit 3 appears **300 times** while writing the integers from 1 to 1000.