The Chezy coefficient (C) is a dimensionless constant used in open channel flow calculations to determine the flow velocity. It is named after the French engineer Antoine Chezy, who developed the Chezy formula.

The Chezy coefficient can be represented in the [MLT] notation system, where M represents mass, L represents length, and T represents time. Let's analyze the dimensions of each option to determine the correct answer:

a) Lb: This option represents length raised to the power of b. However, the Chezy coefficient is dimensionless, so it cannot have a length dimension. Therefore, this option is incorrect.

b) M0 L0 T0: This option represents mass raised to the power of 0, length raised to the power of 0, and time raised to the power of 0. This implies that all the dimensions are raised to the power of 0, which means they are absent. Therefore, this option is incorrect.

c) L1/2 T-1: This option represents length raised to the power of 1/2 and time raised to the power of -1. In the Chezy formula, the Chezy coefficient is typically represented as C = (g * R)^(1/2) / n, where g is the acceleration due to gravity, R is the hydraulic radius, and n is the Manning's roughness coefficient. The dimensions of g * R are L^3/T^2, and dividing it by n would cancel out the dimensions of n. Therefore, the dimensions of the Chezy coefficient are indeed L^(1/2) T^(-1), making this option correct.

d) L-1/4 T2: This option represents length raised to the power of -1/4 and time raised to the power of 2. The Chezy coefficient does not have a length dimension, and its time dimension is -1. Therefore, this option is incorrect.

In conclusion, the correct answer is option 'C' because it correctly represents the dimensions of the Chezy coefficient in the [MLT] notation system as L^(1/2) T^(-1).