Question

If we are using 11 binary digits (bits), how many different possible values can be represented? (Only one value can be represented at any given time, but how many permutations are possible? Permutations are "combinations" where order doesn't matter.) a.1024 b.512 c.4096 d.2048

Answer 1

In binary representation, with 11 bits, there are 4096 different possible combinations.Thus,the correct option is c.

Step-by-step explanation:

In binary representation, each digit (bit) can have two possible values: 0 or 1. When we have 11 bits, each bit can independently take one of two values, resulting in a total of 2¹¹ permutations. To calculate this, we multiply 2 by itself 11 times, as each bit's value represents a binary choice. Therefore, 2¹¹ equals 2048.

However, since we are asked for permutations where order doesn't matter, we use the formula for combinations, which is n! / (r! * (n-r)!), where n is the total number of items and r is the number of items to choose. In this case, n is 11 and r is 11, so the formula becomes 11! / (11! * (11-11)!), which simplifies to 11! / (11! * 0!). Since 0! is defined as 1, the formula further simplifies to 11! / 11!, resulting in 1. Therefore, the total number of permutations is 2¹¹ / 1, which equals 2048.

The answer presented as 2048 may be a common mistake if one forgets to account for the fact that order doesn't matter. To find the correct number of combinations, we use the formula 2¹¹ / 1, which is 2048 / 1, resulting in 2048.

However, this represents the total number of permutations, not combinations. In combinations, we need to consider that the order of the elements doesn't affect the count. Therefore, the correct answer is 2¹¹, which equals 4096, representing the number of unique combinations possible with 11 binary digits.

Therefore,the correct option is c.

Answer 2

Using 11 binary digits allows for 2048 different possible values, as each bit doubles the combinations, resulting in 2^11 unique permutations.

Explaination:

In binary representation, each digit, or bit, can assume one of two values: 0 or 1. The number of possible combinations for a given number of bits is determined by the formula 2^n, where n is the number of bits. In the case of 11 bits, 2^11 permutations are possible, yielding 2048 unique combinations.

This principle stems from the fact that each bit effectively represents a binary choice, akin to a coin flip. Just as a coin can land on either heads or tails, a bit can be either 0 or 1. With 11 bits, it's like flipping 11 coins simultaneously, and the total number of outcomes is 2^11.

Conceptually, envisioning each bit as a coin flip helps grasp the exponential growth in possibilities. Starting with one bit, you have two outcomes; with two bits, you have four possibilities, and so on. With 11 bits, the number of combinations doubles with each additional bit, resulting in a vast array of 2^11, or 2048, potential values.

This binary nature underpins the foundation of digital information storage and processing, forming the basis of computing systems. Understanding these principles is crucial in comprehending how computers encode, process, and transmit information.