Root mean square speed (c) is greater than most probable speed (a) and average speed (b) for identical conditions and the same gas.

Explanation:

1. Most Probable Speed:
The most probable speed of a gas molecule is the speed at which the highest number of molecules in the gas have that particular speed. It is represented by the symbol Vp. In a gas sample, the distribution of molecular speeds follows a Maxwell-Boltzmann distribution, which is a bell-shaped curve. The peak of this curve represents the most probable speed.

2. Average Speed:
The average speed of gas molecules in a sample is the arithmetic mean of the speeds of all the molecules. It is represented by the symbol Vavg. To calculate the average speed, the speeds of all the molecules are summed up and divided by the total number of molecules.

3. Root Mean Square Speed:
The root mean square speed of gas molecules is the square root of the average of the squares of the speeds of all the molecules. It is represented by the symbol Vrms. It is calculated by taking the square root of the average of the squares of the individual speeds of all the gas molecules.

Now, let's compare the three speeds for identical conditions and the same gas:

- Most Probable Speed (Vp):
The most probable speed occurs at the peak of the Maxwell-Boltzmann distribution. It represents the speed at which the highest number of gas molecules have that particular speed. The most probable speed is always lower than the average speed and the root mean square speed.

- Average Speed (Vavg):
The average speed is the arithmetic mean of the speeds of all the gas molecules in the sample. It considers all the speeds in the distribution and is always greater than the most probable speed but smaller than the root mean square speed.

- Root Mean Square Speed (Vrms):
The root mean square speed takes into account the squares of the individual speeds of all the gas molecules. It represents the average speed of the gas molecules considering their kinetic energies. The root mean square speed is always greater than both the most probable speed and the average speed.

Therefore, the correct answer is option 'C', which states that the root mean square speed is greater for identical conditions and the same gas.