Number of Discs on the Driving Shaft = 6
Number of Discs on the Driven Shaft = 5

To find the number of pairs of contact surfaces, we need to consider the number of possible combinations between the discs on the driving shaft and the driven shaft.

1. Determine the number of combinations between the discs on the driving shaft and the discs on the driven shaft:
- We can choose one disc from the driving shaft and one disc from the driven shaft to form a pair of contact surfaces.
- Since there are 6 discs on the driving shaft and 5 discs on the driven shaft, the number of possible combinations is 6 * 5 = 30.

2. However, we need to consider that each pair of contact surfaces consists of two discs, one from the driving shaft and one from the driven shaft.
- For example, if we select disc A from the driving shaft and disc X from the driven shaft, it is the same as selecting disc X from the driving shaft and disc A from the driven shaft. These two combinations represent the same pair of contact surfaces.
- Therefore, we need to divide the total number of combinations by 2 to account for this repetition.
- Dividing 30 by 2, we get 15.

3. Finally, we need to consider that each pair of contact surfaces has two contact surfaces, one on each side of the disc.
- For example, if we consider disc A from the driving shaft and disc X from the driven shaft, there will be a contact surface on one side of disc A and another contact surface on one side of disc X.
- Therefore, the number of pairs of contact surfaces will be twice the number of pairs of discs.
- Multiplying 15 by 2, we get 30.

Therefore, the number of pairs of contact surfaces between the discs on the driving shaft and the driven shaft is 30.