QB_PH_SEC8_012: Digital Electronics

How many 2-to-1 multiplexers are required to construct a single 8-to-1 multiplexer without any external logic gates?

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Multiplexer Tree Architecture**: An 8-to-1 multiplexer selects 1 of 8 data lines using 3 select lines ($S_2, S_1, S_0$). 2. **Cascaded Levels**: - **Level 1 (controlled by $S_0$)**: Groups 8 inputs into pairs: requires $\frac{8}{2} = 4$ multiplexers (produces 4 intermediate outputs). - **Level 2 (controlled by $S_1$)**: Groups the 4 intermediate lines into pairs: requires $\frac{4}{2} = 2$ multiplexers (produces 2 intermediate outputs). - **Level 3 (controlled by $S_2$)**: Selects between the remaining 2 lines: requires $\frac{2}{2} = 1$ multiplexer (produces 1 final output). 3. **Summation**: $N_{\text{total}} = 4 + 2 + 1 = 7$ 4. **Official Answer**: 7.