The total number of integers between 200 and 400, each of which either begins with 3 or ends with 3 or both, is

Options

A. 10
B. 100
C. 110
D. 120

Answer

To solve the problem of finding the total number of integers between 200 and 400 that either begin with 3 or end with 3 (or both), we can break it down into clear steps.

### Step 1: Identify the range
We are looking for integers between 200 and 400. This means we will consider the integers from 201 to 399.

### Step 2: Count numbers that begin with 3
The integers that begin with 3 in this range are:
- 300, 301, 302, 303, 304, 305, 306, 307, 308, 309
- 310, 311, 312, 313, 314, 315, 316, 317, 318, 319
- 320, 321, 322, 323, 324, 325, 326, 327, 328, 329
- 330, 331, 332, 333, 334, 335, 336, 337, 338, 339
- 340, 341, 342, 343, 344, 345, 346, 347, 348, 349
- 350, 351, 352, 353, 354, 355, 356, 357, 358, 359
- 360, 361, 362, 363, 364, 365, 366, 367, 368, 369
- 370, 371, 372, 373, 374, 375, 376, 377, 378, 379
- 380, 381, 382, 383, 384, 385, 386, 387, 388, 389
- 390, 391, 392, 393, 394, 395, 396, 397, 398, 399

Counting these, we find there are **100 integers** that begin with 3.

### Step 3: Count numbers that end with 3
Next, we count the integers between 200 and 400 that end with 3:
- 203, 213, 223, 233, 243, 253, 263, 273, 283, 293
- 303, 313, 323, 333, 343, 353, 363, 373, 383, 393

Counting these, we find there are **20 integers** that end with 3.

### Step 4: Count numbers that both begin and end with 3
Now, we need to check if there are any numbers that both begin with 3 and end with 3. The only number in our range that meets this criterion is:
- 303, 313, 323, 333, 343, 353, 363, 373, 383, 393

Counting these, we find there are **10 integers** that both begin and end with 3.

### Step 5: Apply the principle of inclusion-exclusion
To find the total number of integers that either begin with 3 or end with 3, we use the principle of inclusion-exclusion:
\[ \text{Total} = (\text{Count of numbers starting with 3}) + (\text{Count of numbers ending with 3}) - (\text{Count of numbers starting and ending with 3}) \]
\[ \text{Total} = 100 + 20 - 10 = 110 \]

### Conclusion
The total number of integers between 200 and 400 that either begin with 3 or end with 3 (or both) is **110**.

---