Let x, y, and z represent the number of singles, fours, and sixes scored by the batsman, respectively.

The equation x + 4y + 6z = 25 represents the total runs scored by the batsman. wherein x, y, z ≥ 0

Case 1. If no sixes have been hit (z = 0),

•   x + 4y = 25.

•   There are 7 possible combinations of values for x and y, when no sixes have been hit.

•   (x, y) = (1, 6), (5, 5), (9, 4), (13, 3), (17, 2), (21, 1), (25, 0).

Case 2. If one six has been hit (z=1),

•   x + 4y = 19.

•   There are 5 possible combinations of values for x and y when one six has been hit.

•   (x, y) = (3, 4), (7, 3), (11, 2), (15, 1), (19, 0)

Case 3. If two sixes have been hit (z=2),

•   x + 4y = 13.

•   There are 4 possible combinations of values for x and y when two sixes have been hit.

•   (x, y) = (1, 3), (5, 2), (9, 1), (13, 0)

Case 4. If three sixes have been hit (z=3),

•   x + 4y = 7

•   So, the possible values of (x, y) may be (3, 1), (7, 0), i.e. 2 possible ways.

•   There are 2 possible combinations of values for x and y when three sixes have been hit.

•   (x, y) = (3, 1), (7, 0)

Case 5. If four sixes have been hit (z=4),

•   x + 4y = 1

•   There is 1 possible combination of values for x and y when four sixes have been hit.

•   (x, y) = (1, 0)

Total Ways

•   Total number of possible ways = 7 + 5 + 4 + 2 + 1 = 19

•   Therefore, the correct answer is option (b).