Decoding Binary 1101 using a 4-input AND gate and one inverter

To decode binary 1101, we need to use a combination of logic gates that can convert the binary input into a corresponding output. Let's analyze each option and determine which one is the correct choice:

a) One 4-input AND gate:
- An AND gate takes multiple inputs and produces an output that is HIGH (logic 1) only when all of its inputs are HIGH.
- In this case, a 4-input AND gate would require all four inputs to be HIGH in order to produce a HIGH output.
- Since the binary input 1101 has one input as LOW (0), this option is not suitable for decoding this binary.

b) One 4-input AND gate, one inverter:
- An inverter (also known as a NOT gate) produces the opposite logic level of its input. If the input is HIGH, the output is LOW, and vice versa.
- By using a 4-input AND gate and one inverter, we can achieve the desired decoding of binary 1101.
- The AND gate will check if all inputs are HIGH, and the inverter will invert the output of the AND gate, making it HIGH when the AND gate output is LOW.
- In this case, the 4-input AND gate will produce a LOW output since one input is LOW (0), and the inverter will invert this LOW output to HIGH.
- Therefore, this combination of logic gates can successfully decode binary 1101.

c) One 4-input AND gate, one OR gate:
- An OR gate produces a HIGH output when any of its inputs are HIGH.
- Combining a 4-input AND gate and an OR gate will not produce the desired decoding of binary 1101.
- The AND gate will check if all inputs are HIGH, and if any input is LOW, the AND gate output will be LOW.
- The OR gate will then produce a HIGH output since at least one input is LOW.
- Therefore, this combination of logic gates is not suitable for decoding binary 1101.

d) One 4-input NAND gate, one inverter:
- A NAND gate produces a LOW output only when all of its inputs are HIGH.
- Combining a 4-input NAND gate and an inverter will not produce the desired decoding of binary 1101.
- The NAND gate will produce a HIGH output since one input is LOW (0).
- The inverter will invert this HIGH output to LOW, which is not the desired output for decoding binary 1101.
- Therefore, this combination of logic gates is not suitable for decoding binary 1101.

Conclusion:
Based on the analysis above, the correct combination of logic gates to decode binary 1101 is option 'b' - one 4-input AND gate and one inverter. This combination ensures that the output is HIGH when the binary input is 1101 and LOW for all other inputs.