Convert the IPv4 address 192.168.10.15 from dotted-decimal notation to its 32-bit binary representation.

Introduction / Context:
In computer networking, IPv4 addresses are often written in dotted-decimal form for human readability, but routers and hosts work with binary values. Being able to convert between dotted-decimal and binary notation is a fundamental skill for understanding subnetting, masks, and routing behavior in TCP/IP networks.

Given Data / Assumptions:

• Given IPv4 address: 192.168.10.15.
• An IPv4 address consists of four octets, each 8 bits.
• Each octet in decimal must be converted to an 8-bit binary number.
• We maintain leading zeros in each octet to show all 8 bits.

Concept / Approach:
The approach is to convert each octet from decimal to binary using powers of two. Each octet ranges from 0 to 255 and is represented by bits with place values 128, 64, 32, 16, 8, 4, 2, and 1. For each octet, we repeatedly subtract the largest power of two that fits and mark a 1 in that position, leaving 0 where the power of two does not fit. We then join the octets with dots to represent the full 32-bit binary address.

Step-by-Step Solution:
1. Convert 192 to binary: 192 = 128 + 64, so bits are 11000000.2. Convert 168 to binary: 168 = 128 + 32 + 8, so bits are 10101000.3. Convert 10 to binary: 10 = 8 + 2, so bits are 00001010.4. Convert 15 to binary: 15 = 8 + 4 + 2 + 1, so bits are 00001111.5. Combine the four 8-bit groups with dots: 11000000.10101000.00001010.00001111.

Verification / Alternative check:
You can verify correctness by converting each binary octet back to decimal. For example, 11000000 is 128 + 64 = 192, 10101000 is 128 + 32 + 8 = 168, 00001010 is 8 + 2 = 10, and 00001111 is 8 + 4 + 2 + 1 = 15. This round-trip check confirms that the binary representation matches the original dotted-decimal address.

Why Other Options Are Wrong:
Option B swaps the last two octets, giving the wrong host portion. Option C changes one bit in the second octet, altering 168 to a different decimal value. Option D changes bits in the last octet, turning 15 into another number. Any such change results in a different IP address and therefore an incorrect conversion.

Common Pitfalls:
Common mistakes include forgetting to pad each octet to 8 bits with leading zeros, mixing up octet order, or miscalculating the contribution of each power of two. Another pitfall is writing binary without dots, which makes it harder to see the four-octet structure of the IPv4 address.

Final Answer:
11000000.10101000.00001010.00001111