Question

How many 421 multiplexers are required to implement a 16 to 1 multiplexer?

Answer 1

To implement a 16-to-1 multiplexer using 4-to-1 multiplexers (421 multiplexers), we need to carefully consider the way multiplexers work and how we can combine them to create the desired larger multiplexer.

### Understanding the Multiplexer Structure

1. **4-to-1 Multiplexer (421 Mux)**: A 4-to-1 multiplexer has 4 input lines, 1 output line, and 2 select lines. This means that with 2 select lines, it can choose one of 4 inputs to pass to the output.

2. **16-to-1 Multiplexer**: A 16-to-1 multiplexer has 16 input lines, 1 output line, and 4 select lines. The 4 select lines allow the multiplexer to select one of the 16 inputs.

### Step-by-Step Approach to Implementing a 16-to-1 Mux

We will use 4-to-1 multiplexers to build the 16-to-1 multiplexer. Here’s how we can do it:

#### Level 1: First Stage (Combine the Inputs)
- Since each 4-to-1 multiplexer can handle 4 inputs, we will need **4 multiplexers** to handle the 16 input lines. Each of the 4 multiplexers will select one of the 4 inputs from the 16 inputs.
- Thus, in the first stage, we will have 4 multiplexers, each taking 4 inputs and selecting 1 output. This gives us 4 outputs from these multiplexers.

#### Level 2: Second Stage (Select from the Outputs of the First Stage)
- Now, we need to select from the 4 outputs of the first stage.
- We need a multiplexer that can select 1 output from these 4 outputs, so we use **1 more 4-to-1 multiplexer** to handle this. The output of this second-stage multiplexer is the final output of the 16-to-1 multiplexer.

#### Total Number of Multiplexers
- In the first stage, we used **4 multiplexers** (to handle the 16 inputs).
- In the second stage, we used **1 multiplexer** (to select from the 4 outputs of the first stage).

So, the total number of **4-to-1 multiplexers** required to implement a 16-to-1 multiplexer is:

\[
\boxed{5 \text{ multiplexers}}.
\]

### Explanation of the Number of Multiplexers
- The first stage handles the initial selection from the 16 inputs (using 4 multiplexers).
- The second stage consolidates the 4 outputs from the first stage (using 1 multiplexer).
- This efficient use of 4-to-1 multiplexers requires a total of 5 multiplexers for the 16-to-1 implementation.