Question
A store uses binary numbers to assign a unique binary sequence to each item in its inventory. What is the minimum number of bits required for each binary sequence if the store has between 75 and 100 items in its inventory?
5
Option A,5
6
Option B,6
7
Option C,7
8
5
Option A,5
6
Option B,6
7
Option C,7
8
Answers
GPT-4o mini
To determine the minimum number of bits required to uniquely represent a certain number of items in a binary system, we can use the formula:
\[
N = 2^n
\]
where \( N \) is the number of unique combinations (or items in this case) and \( n \) is the number of bits.
1. For \( n = 5 \):
\[
2^5 = 32 \quad \text{(not enough for 75 to 100 items)}
\]
2. For \( n = 6 \):
\[
2^6 = 64 \quad \text{(not enough for 75 to 100 items)}
\]
3. For \( n = 7 \):
\[
2^7 = 128 \quad \text{(sufficient for 75 to 100 items)}
\]
4. For \( n = 8 \):
\[
2^8 = 256 \quad \text{(also sufficient but overkill)}
\]
Since the store has between 75 and 100 items, the minimum number of bits required is \( n = 7 \) because \( 2^7 = 128 \) is the smallest power of 2 that can accommodate at least 75 items and up to 100.
Thus, the minimum number of bits required is:
**Option C: 7**
\[
N = 2^n
\]
where \( N \) is the number of unique combinations (or items in this case) and \( n \) is the number of bits.
1. For \( n = 5 \):
\[
2^5 = 32 \quad \text{(not enough for 75 to 100 items)}
\]
2. For \( n = 6 \):
\[
2^6 = 64 \quad \text{(not enough for 75 to 100 items)}
\]
3. For \( n = 7 \):
\[
2^7 = 128 \quad \text{(sufficient for 75 to 100 items)}
\]
4. For \( n = 8 \):
\[
2^8 = 256 \quad \text{(also sufficient but overkill)}
\]
Since the store has between 75 and 100 items, the minimum number of bits required is \( n = 7 \) because \( 2^7 = 128 \) is the smallest power of 2 that can accommodate at least 75 items and up to 100.
Thus, the minimum number of bits required is:
**Option C: 7**