Question

A Decoder with n inputs has the following number of outputs:

• A. n
• B. \( 2^n \)
• C. \( n^2 \)
• D. \( \log n \)

Hint:

A decoder can also be used as a demultiplexer (DEMUX) if the enable line is treated as the single data input.
The input-to-output relationship is always \( n \)-to-\( 2^n \).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The question asks for the mathematical relationship between the number of input lines and the output lines in a standard binary decoder.
A decoder is a combinational logic circuit that translates an \( n \)-bit binary code into unique active outputs.

Step 2: Key Formula or Approach:
For an \( n \)-bit input combination, there are exactly \( 2^n \) possible binary states.
A standard binary decoder is designed such that for each input combination, exactly one of the outputs is activated (made high or low depending on active-high or active-low logic).
Therefore, the decoder has \( n \) input lines and \( 2^n \) output lines.

Step 3: Detailed Explanation:
Let us look at some common configurations of decoders to verify this relationship:

• 2-to-4 Decoder: Here, \( n = 2 \). The number of outputs is \( 2^2 = 4 \).
The binary inputs \( 00, 01, 10, 11 \) select outputs \( Y_0, Y_1, Y_2, Y_3 \) respectively.

• 3-to-8 Decoder: Here, \( n = 3 \). The number of outputs is \( 2^3 = 8 \).
The inputs address outputs ranging from \( Y_0 \) to \( Y_7 \).

• 4-to-16 Decoder: Here, \( n = 4 \). The number of outputs is \( 2^4 = 16 \).

• Functionality: Decoders are highly useful in memory addressing, where the CPU uses address lines as inputs to select a specific memory chip or register location out of \( 2^n \) unique addresses.

Step 4: Final Answer:
A decoder with \( n \) inputs has \( 2^n \) outputs.