Discharge through a centrifugal pump can be calculated using the formula Q = πDvw, where Q is the discharge, D is the diameter of the impeller at the inlet, v is the velocity of flow at the inlet, and w is the width of the impeller at the inlet.

To understand why this formula is used, let's break it down:

1. Diameter of the impeller at the inlet (D):
- The diameter of the impeller refers to the size of the rotating component of the pump.
- It determines the amount of fluid that can be carried by the pump.
- A larger impeller diameter allows for a greater volume of fluid to pass through the pump.

2. Velocity of flow at the inlet (v):
- The velocity of flow refers to the speed at which the fluid enters the pump.
- It is typically measured in meters per second (m/s).
- The velocity of flow affects the amount of fluid that can be carried by the pump.
- A higher velocity of flow results in a greater volume of fluid being discharged.

3. Width of the impeller at the inlet (w):
- The width of the impeller refers to the dimension of the impeller in the direction perpendicular to its diameter.
- It determines the cross-sectional area available for fluid flow.
- A larger impeller width allows for a greater volume of fluid to pass through the pump.

4. Discharge (Q):
- The discharge refers to the volume of fluid that is pumped per unit of time.
- It is typically measured in cubic meters per second (m³/s) or liters per second (L/s).
- The discharge is calculated by multiplying the cross-sectional area (πDw) with the velocity of flow (v).

Now, let's apply the formula to the given options:

a) πDVf:
- This option includes the correct factors (π, D, v), but it is missing the impeller width (w).
- Therefore, it is not the correct answer.

b) DVfw:
- This option includes the correct factors (D, v, w), but it is missing the factor of π.
- The factor of π is necessary to calculate the cross-sectional area of the impeller.
- Therefore, it is not the correct answer.

c) πDVfw:
- This option includes all the correct factors (π, D, v, w).
- The cross-sectional area (πDw) is multiplied by the velocity of flow (v) to calculate the discharge (Q).
- Therefore, this is the correct answer.

d) πDw:
- This option includes the correct factors (π, D, w), but it is missing the factor of v.
- The velocity of flow (v) is necessary to calculate the discharge (Q).
- Therefore, it is not the correct answer.

In conclusion, the correct option for the discharge through a centrifugal pump is c) πDVfw.