The major functioning responsibility of a multiplexing combinational circuit is the generation of a selected path between multiple sources and a single destination. Let's break down this answer in detail:

1. What is a multiplexing combinational circuit?
A multiplexing combinational circuit is a digital circuit that allows multiple input signals to be transmitted over a single transmission line or channel. It uses a combination of logic gates to select and route the desired input signal to the output.

2. Understanding the options:
a) Decoding the binary information:
Decoding refers to the process of converting a binary code into a more meaningful form. While decoding can be a part of the multiplexing process, it is not the major functioning responsibility of the circuit.

b) Generation of all minterms in an output function with OR-gate:
A minterm is a product term in Boolean algebra that represents a specific combination of input variables. While generating minterms can be a part of the multiplexing process, it is not the major functioning responsibility of the circuit.

c) Generation of selected path between multiple sources and a single destination:
This is the correct answer. The main purpose of a multiplexing combinational circuit is to select and route the desired input signal to a single output destination. It allows multiple sources to share a common transmission line or channel, enabling efficient communication.

d) Encoding of binary information:
Encoding refers to the process of converting meaningful data into a coded form. While encoding can be a part of the multiplexing process, it is not the major functioning responsibility of the circuit.

3. Importance of selecting a path:
In many applications, especially in communication systems, there is a need to transmit multiple signals over limited resources. Multiplexing allows efficient utilization of these resources by selecting and routing the desired signals to their respective destinations. By choosing the appropriate path, the multiplexing combinational circuit ensures that the desired signal reaches the destination without interference or loss.

4. How the circuit works:
The multiplexing combinational circuit uses logic gates, such as AND gates and OR gates, to select the desired input signal based on control signals. These control signals determine which input signal is routed to the output. By manipulating the control signals, different input signals can be selected and transmitted at different times, effectively sharing the transmission line or channel.

In conclusion, the major functioning responsibility of a multiplexing combinational circuit is to generate a selected path between multiple sources and a single destination. It allows efficient utilization of resources and enables the transmission of multiple signals over a common transmission line or channel.