Correct Option (1)

The problem requires identifying the count of numbers between 100 and 300 that either begin with the digit 2 or end with the digit 2.

First, consider numbers ending with 2:

Numbers in the range 101-199 that end with 2 are 102, 112, 122, 132, 142, 152, 162, 172, 182, 192. There are 10 such numbers.
Numbers in the range 200-299 that end with 2 are 202, 212, 222, 232, 242, 252, 262, 272, 282, 292. There are 10 such numbers.
The total count of numbers ending with 2 is 10 + 10 = 20.

Next, consider numbers beginning with 2:

Numbers between 100 and 300 that begin with 2 fall in the range 200-299.
The count of these numbers is 299 - 200 + 1 = 100.

To determine the total unique count, we must account for numbers that satisfy both conditions (i.e., begin with 2 AND end with 2) to avoid double-counting. These are the numbers from 200-299 that also end with 2:

The numbers that begin with 2 and end with 2 are 202, 212, 222, 232, 242, 252, 262, 272, 282, 292. There are 10 such numbers.

Using the Principle of Inclusion-Exclusion:

Total = (Numbers ending with 2) + (Numbers beginning with 2) - (Numbers beginning with 2 AND ending with 2)

Total = 20 + 100 - 10 = 110.

Alternatively, by summing distinct sets:

Numbers ending with 2 (from both 100s and 200s ranges) = 20.
Numbers beginning with 2 but NOT ending with 2 = (Total numbers beginning with 2) - (Numbers beginning with 2 AND ending with 2) = 100 - 10 = 90.
Total numbers = (Numbers ending with 2) + (Numbers beginning with 2 but not ending with 2) = 20 + 90 = 110.

Incorrect Options:

Options 2, 3, and 4 are incorrect because the systematic enumeration and application of counting principles demonstrate that there are precisely 110 such numbers between 100 and 300.