Miller successfully demonstrated the formation of amino acids by electric discharge in a mixture of hydrogen, ammonia, water, and methane.

Explanation:
1. Miller's Experiment:
Miller conducted an experiment in 1953 to mimic the conditions of early Earth's atmosphere and test the hypothesis that organic molecules could have been formed under those conditions. He set up a closed system containing a mixture of gases that were thought to be present in the early atmosphere, such as hydrogen (H2), ammonia (NH3), water vapor (H2O), and methane (CH4). This mixture represented the reducing atmosphere that was believed to exist at that time.

2. Electric Discharge:
Miller then subjected this mixture to electric discharge, simulating lightning or energy input from volcanic activity. The electric discharge provided the necessary energy to drive chemical reactions and simulate the energy input from natural sources.

3. Formation of Amino Acids:
After running the electric discharge for a week, Miller analyzed the contents of the system and found the presence of several organic compounds, including amino acids. Amino acids are the building blocks of proteins, which are essential for life. The formation of amino acids in Miller's experiment provided evidence that the basic building blocks of life could have been synthesized under the conditions present on early Earth.

4. Role of Methane:
Among the gases used in Miller's experiment, methane (CH4) played a crucial role. Methane is a simple organic molecule that contains carbon and hydrogen atoms. It can react with other molecules under the influence of energy to form more complex organic compounds, such as amino acids.

5. Importance of Hydrogen and Ammonia:
Hydrogen (H2) and ammonia (NH3) were also present in the mixture. Hydrogen provided a reducing environment, meaning it had a surplus of electrons, which facilitated the formation of organic compounds. Ammonia acted as a source of nitrogen, an essential element for the formation of amino acids.

6. Exclusion of Ether, Ethane, and Carbon Dioxide:
The correct answer is option C, methane (CH4), because it is the only gas among the given options that was used in Miller's experiment and contributed to the formation of amino acids. Ether, ethane, and carbon dioxide were not included in Miller's original experiment and therefore did not play a role in the formation of amino acids in his study.

In conclusion, Miller successfully demonstrated the formation of amino acids by electric discharge in a mixture of hydrogen, ammonia, water, and methane. This experiment provided evidence that the basic building blocks of life could have been synthesized under the conditions present on early Earth.