Question

A group of eight distinct elements can be represented by a ______ code.

• A. Three-bit
• B. Two-bit
• C. Four-bit
• D. Eight-bit

Hint:

Remember: \[ \text{Number of combinations}=2^n \] \[ 8=2^3 \] Therefore 8 distinct elements require a 3-bit code.

Answer 1

The Correct Option is A

Concept: If a code consists of \(n\) bits, the total number of distinct combinations possible is: \[ 2^n \] To represent a given number of elements, we must choose the smallest value of \(n\) such that: \[ 2^n \geq \text{Number of Elements} \]

Step 1: Identify the number of elements.
The question states that there are: \[ 8 \] distinct elements.

Step 2: Determine the number of bits required.
Check powers of 2: \[ 2^1=2 \] \[ 2^2=4 \] \[ 2^3=8 \] \[ 2^4=16 \] We observe that: \[ 2^3=8 \] Therefore exactly three bits can represent eight distinct elements.

Step 3: Verify using binary combinations.
Three bits generate: \[ 000,\;001,\;010,\;011,\;100,\;101,\;110,\;111 \] Total combinations: \[ 8 \] Hence a three-bit code is sufficient. \[ \boxed{\text{Three-bit Code}} \] Therefore, option (A) is correct.