How many 3-to-8 line decoders are required for a 1-of-32 decoder?

Concept

Decoder expansion: a small n-to-2n decoder can be combined to build a larger one. The number of small decoders needed to supply the outputs equals the big decoder's total outputs divided by one small decoder's outputs. A 3-to-8 decoder has 3 select inputs and 23 = 8 output lines.

Application

1. A 1-of-32 decoder needs 32 distinct output lines (it activates one of 32 lines).

2. Each 3-to-8 decoder supplies 8 output lines.

3. Number of 3-to-8 decoders to cover all outputs = 32 / 8 = 4.

4. A 5-bit address (25 = 32) splits as the lower 3 bits feeding every 3-to-8 decoder identically and the upper 2 bits choosing which one of the four decoders is active.

5. The remaining 2 address bits are consumed by the ENABLE inputs already built into each 3-to-8 decoder (e.g. a 74138's enable pins) -- the two bits, in their true and inverted forms (from a single inverter), are wired across the four units' enable pins so that exactly one decoder is active for each of the 4 combinations -- so selecting among the four decoders needs no fifth decoder of any size, only one basic inverter gate.

Cross-check

Four decoders give 4 x 8 = 32 output lines, exactly matching the requirement, with no line wasted. The 2 extra address bits are absorbed by the four decoders' own built-in enable pins (needing just one inverter), not by another decoder unit -- this matches the standard manufacturer expansion scheme (e.g. the 74138 datasheet's 1-of-32 application, which uses exactly four 3-to-8 ICs plus a single inverter). That is why only four 3-to-8 decoders are counted.

Result: 4 three-to-eight decoders are required.